#6357008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264204);