#622121

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

1632 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

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, 529317);