#696637

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

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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,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, 603833);