#6792405

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

1885 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)(5,19)(6,9)(7,14)(8,17)(10,16)(12,22)(13,24)(18,29)(20,21)(23,30)(25,31)(26,32)(27,28);(1,3)(2,7)(4,12)(5,20)(6,13)(8,22)(9,21)(10,24)(11,19)(14,16)(15,28)(17,31)(18,26)(23,32)(25,29)(27,30);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,22)(13,21)(15,17)(16,19)(18,23)(20,24)(25,27)(26,32)(28,31)(29,30);(1,4,16,30,6,18,5,17)(2,8,19,29,9,23,10,15)(3,12,14,27,13,26,20,31)(7,22,11,25,21,32,24,28)

Polycyclic presentation:

2^9|e^2=fhi,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, 6699601);