#158790

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

2472 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=fi,d^2=fgh,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, 65986);