#1411911

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
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

1558 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)(9,10,14,16)(11,12,15,13)(25,26)(27,32)(28,37)(29,34)(30,31)(33,39)(35,38)(36,40);(9,11,14,15)(10,12,16,13)(18,21)(22,24)(26,31)(27,35)(33,37)(34,36);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,35)(26,32,31,38)(28,34,39,36)(29,33,40,37);(9,12,14,13)(10,11,16,15)(17,19,20,23)(18,22,21,24)(25,28)(26,33)(27,36)(29,32)(30,39)(31,37)(34,35)(38,40);(1,3,4,8)(2,5,6,7)(9,13,14,12)(10,15,16,11)(25,29)(26,34)(27,33)(28,38)(30,40)(31,36)(32,39)(35,37)

Polycyclic presentation:

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

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, 1319107);