#727100

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=fghi,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, 634296);