#7624867

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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

1335 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)(10,17)(11,20)(13,16)(15,22)(18,32)(19,34)(21,33)(23,35)(24,38)(25,37)(26,36)(28,31)(29,30)(39,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,21)(12,16)(13,27)(14,17)(15,19)(18,25)(20,32)(22,28)(23,24)(26,39)(29,34)(30,31)(33,37)(35,36)(38,40);(9,11,22,38,16,26,30,37,14,25,29,36,13,24,15,20)(10,18,28,35,12,23,31,32,17,21,34,40,27,39,19,33);(3,8)(5,7)(9,12)(10,13)(11,23)(14,27)(15,28)(16,17)(18,24)(19,29)(20,35)(21,26)(22,31)(25,39)(30,34)(32,38)(33,36)(37,40);(2,6)(5,7)(9,13,14,16)(10,12,17,27)(11,24,25,26)(15,29,30,22)(18,23,21,39)(19,28,31,34)(20,36,37,38)(32,40,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315178, #72315824, #72315829
maximal normal subgroup of	#72315178, #72315824, #72315829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532063);