#7623178

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
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	#60648	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1624 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)(3,6)(4,5)(7,8)(9,10,17,11)(12,14,25,32)(13,19,28,23)(15,24,18,20)(16,26,22,21)(27,29,38,40)(30,37,33,34)(31,39,36,35);(2,5)(6,8)(9,11,15,25,16,26,14,24,17,10,18,12,22,21,32,20)(13,23,30,38,31,39,29,37,28,19,33,27,36,35,40,34);(1,3,4,7)(2,6,5,8)(9,12,14,11,22,24,15,21,17,25,32,10,16,20,18,26)(13,27,29,23,36,37,30,35,28,38,40,19,31,34,33,39);(9,13)(10,19)(11,23)(12,27)(14,29)(15,30)(16,31)(17,28)(18,33)(20,34)(21,35)(22,36)(24,37)(25,38)(26,39)(32,40);(13,28)(19,23)(27,38)(29,40)(30,33)(31,36)(34,37)(35,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530374);