#950289

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1458 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 857485);