#696003

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1628 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 603199);