#6357840

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357820
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

1996 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(18,19)(21,22)(24,27)(25,26)(28,32)(30,31);(2,6)(7,14)(8,13)(9,15)(17,18,22,25,23,26,21,19)(20,24,31,28,29,32,30,27);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,19,21,26,23,25,22,18)(20,28,30,24,29,27,31,32);(3,10)(7,14)(9,15)(11,16)(17,20)(18,24)(19,27)(21,30)(22,31)(23,29)(25,28)(26,32);(20,29)(24,32)(27,28)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265036);