#160358

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
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

2396 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)(9,10,13,11)(12,14,16,15)(17,18)(19,21)(20,23)(22,24);(2,4)(10,11)(12,16)(18,19)(20,24)(25,26,28,30)(27,31,32,29);(5,6)(7,8)(9,11,13,10)(12,14,16,15)(17,19,21,18)(20,22,24,23)(25,27,28,32)(26,29,30,31);(1,2)(3,4)(9,12)(10,14)(11,15)(13,16)(17,20,21,24)(18,22,19,23)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,c^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67554);