#7000522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6907718);