#158403

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=efg,d^2=eh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65599);