#6792404

order	512
nilpotency class	3
isoclinism	isoclinic class of #6792404
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#38904
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6699600);