#6626511

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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)(3,7)(4,6)(5,8)(17,18,23,19)(20,28,30,32)(21,25,22,26)(24,31,27,29);(1,3,5,4)(2,6,8,7)(9,10,14,12)(11,16,13,15)(17,19,23,18)(20,27,30,24)(21,25,22,26)(28,29,32,31);(1,3,5,4)(2,7,8,6)(9,11,14,13)(10,15,12,16)(17,20)(18,24)(19,27)(21,29)(22,31)(23,30)(25,32)(26,28);(1,4,5,3)(2,6,8,7)(9,12,14,10)(11,15,13,16)(17,21,23,22)(18,25,19,26)(20,29,30,31)(24,32,27,28);(9,13,14,11)(10,16,12,15)(17,21,23,22)(18,26,19,25)(20,29,30,31)(24,28,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fh,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6533707);