#982873

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1668 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 890069);