#7624226

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

1553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,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, #72320147, #72320368, …
maximal normal subgroup of	#72287781, #72320147, #72320368, …
maximal quotient of	#72343810

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531422);