#949896

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

1694 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=hi,e^2=f,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, 857092);