#6357999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357957
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

1925 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,23,26)(19,27,29,21)(20,28,25,22)(24,31,32,30);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,28,31,23,29,22,30)(18,24,20,21,26,32,25,27);(17,20,22,18,23,25,28,26)(19,21,30,24,29,27,31,32);(1,3,5,10)(2,8,9,15)(4,11,7,14)(6,12,13,16)(17,21,23,27)(18,19,26,29)(20,31,25,30)(22,24,28,32);(3,10)(8,15)(11,14)(12,16)(19,29)(21,27)(24,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265195);