#7624349

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287781, #72320157, #72320335, …
maximal normal subgroup of	#72287781, #72320157, #72320335, …
maximal quotient of	#72343816

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531545);