#6357727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264923);