#631770

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

1572 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=gi,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 538966);