#1148904

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1526 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)(3,7,8,5)(9,10,13,16)(11,15,12,14)(19,23)(22,24)(26,27)(28,35)(29,37)(38,39);(18,21)(22,24)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,16,15)(17,18,20,21)(19,22,23,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,2)(3,5)(4,6)(7,8)(17,19,20,23)(18,22,21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,3,4,8)(2,5,6,7)(9,12,13,11)(10,15,16,14)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1056100);