#950627

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=f,d^2=i,e^2=fh,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, 857823);