#6357642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357620
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

1990 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=ghi,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^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, 6264838);