#707296

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1578 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=fh,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, 614492);