#72287872

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	#7623257	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2227 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:

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

Polycyclic presentation:

2^10|c^2=h,d^2=gij,e^2=f,f^2=h,g^2=ij,i^2=j,c^a=chj,d^b=dj,d^c=dg,e^b=eh,e^c=ej,e^d=egj,f^d=fj,g^c=gi,g^e=gij,i^c=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318539, #72319144, #72319699
maximal normal subgroup of	#72318539, #72319144, #72319699

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, 3, F.8);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);