#1287165

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1194361);