#72287063

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#10575374	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5076 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:

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

Polycyclic presentation:

2^10|d^2=f,e^2=fgh,f^2=h,g^2=i,h^2=j,i^2=j,c^a=cj,d^a=dj,d^b=dhi,e^d=eg,f^e=fi,g^d=gj,g^e=gi,h^e=hj,i^e=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287099, #72287390, #72319190, …
maximal normal subgroup of	#72287099, #72287390, #72319190, …

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