#158226

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

2328 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=gh,d^2=efg,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, 65422);