#1247500

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=g,d^2=g,e^2=ghi,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, 1154696);