#6572461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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)(17,18,23,21)(19,22,26,24)(20,27,31,28)(25,32,29,30);(2,5)(6,8)(9,10,14,13)(11,16,12,15)(17,19)(18,24)(20,28)(21,22)(23,26)(25,30)(27,31)(29,32);(9,11,14,12)(10,15,13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,32)(26,28);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,15,13,16)(17,21,23,18)(19,22,26,24)(20,29,31,25)(27,30,28,32);(9,13,14,10)(11,16,12,15)(17,22,23,24)(18,26,21,19)(20,30,31,32)(25,28,29,27)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479657);