#931026

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1674 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fghi,d^2=h,e^2=f,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, 838222);