#1126653

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

1440 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=g,e^2=fgh,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, 1033849);