#880832

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=f,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 788028);