#6350940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2637 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258136);