#698206

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

1626 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)(9,10,12,14)(11,13,15,16)(17,18,21,20)(19,23,24,22)(25,26)(27,32)(28,36)(29,34)(30,31)(33,39)(35,38)(37,40);(1,3,5,4)(2,6,8,7)(19,24)(22,23)(26,31)(29,32)(33,36)(38,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27,30,34)(26,32,31,29)(28,37,39,35)(33,38,36,40);(1,2)(3,6)(4,7)(5,8)(17,18,21,20)(19,22,24,23)(25,28,30,39)(26,33,31,36)(27,35,34,37)(29,38,32,40);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,23,24,22)(25,29,30,32)(26,34,31,27)(28,38,39,40)(33,37,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,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, 605402);