#6357631

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

1888 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)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18,24,30)(19,21,27,26)(20,32,31,25)(22,28,23,29);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,23,20,24,27,22,31)(18,25,28,26,30,32,29,21);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(17,20,23,27,24,31,22,19)(18,26,28,32,30,21,29,25);(2,7)(6,13)(8,14)(12,16)(17,21)(18,27)(19,30)(20,29)(22,32)(23,25)(24,26)(28,31);(17,22,24,23)(18,28,30,29)(19,31,27,20)(21,25,26,32)

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^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, 6264827);