#7594822

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=g,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502018);