#6357889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357861
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) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,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, 6265085);