#72287457

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623529	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17336 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,7)(4,12)(8,10)(11,14)(17,18)(19,26)(20,30)(21,28)(22,40)(23,31)(24,32)(25,27)(29,42)(33,39)(34,35)(36,45)(37,47)(38,44)(41,46)(43,48);(1,2,5,10,6,7,3,8)(4,13,11,15,12,16,14,9)(17,19,23,29,25,28,44,45,24,34,43,40,20,33,37,46)(18,26,31,42,27,21,38,36,32,35,48,22,30,39,47,41);(1,3,6,5)(2,8,7,10)(4,14,12,11)(9,16,15,13)(17,20,24,25)(18,27,32,30)(19,33,34,28)(21,35,39,26)(22,41,42,36)(23,37,43,44)(29,46,40,45)(31,38,48,47);(1,4,3,11,6,12,5,14)(2,9,8,13,7,15,10,16)(17,21,20,35,24,39,25,26)(18,28,30,34,32,33,27,19)(22,37,41,43,42,44,36,23)(29,38,45,31,40,47,46,48);(4,14,12,11)(9,16,15,13)(17,22,23,21,25,41,44,35,24,42,43,39,20,36,37,26)(18,29,31,33,27,45,38,19,32,40,48,28,30,46,47,34)

Polycyclic presentation:

2^10|b^2=g,c^2=h,d^2=f,e^2=fghij,f^2=h,g^2=i,i^2=j,b^a=bhj,c^a=cj,d^a=dhj,d^b=dghij,d^c=dhj,e^a=ej,e^b=ehij,e^c=ej,e^d=egj,f^b=fj,f^e=fj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287517, #72318841, #72319144, …
maximal normal subgroup of	#72287517, #72318841, #72319144, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);