#7624863

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=i,e^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=df,e^b=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287059, #72287315, #72287329, …
maximal normal subgroup of	#72287059, #72287315, #72287329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532059);