#7623181

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60647	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1670 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)(3,6)(4,5)(7,8)(10,17)(11,24)(13,18)(14,31)(15,19)(20,32)(21,22)(23,26)(25,38)(28,33)(29,40)(30,34)(35,39)(36,37);(2,5)(6,8)(9,10,18,11,15,22,31,26,16,23,14,21,19,24,13,17)(12,20,33,25,30,37,40,39,27,35,29,36,34,38,28,32);(1,3,4,7)(2,6,5,8)(9,11,13,10,19,17,14,24,16,21,31,23,15,26,18,22)(12,25,28,20,34,32,29,38,27,36,40,35,30,39,33,37);(9,12)(10,20)(11,25)(13,28)(14,29)(15,30)(16,27)(17,32)(18,33)(19,34)(21,36)(22,37)(23,35)(24,38)(26,39)(31,40);(12,27)(20,35)(25,36)(28,40)(29,33)(30,34)(32,39)(37,38)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322758
maximal normal subgroup of	#72322758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530377);