#770927

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,6)(3,8)(11,15)(14,16)(18,22)(23,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(9,10,13,12)(11,14,15,16)(18,22)(23,24)(26,30)(27,33)(32,35)(34,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3,5,8)(2,7,6,4)(17,19,21,20)(18,23,22,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,24,22,23)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

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, 678123);