#6856530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667652
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 ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi

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, 6763726);