#7623997

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
prime radical	#61427	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1297 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531193);