#6908089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,e^d=ei

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, 6815285);