#6627712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534908);