#1249426

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

1522 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)(9,10,13,16)(11,15,12,14)(17,18,21,20)(19,22,24,23)(25,26)(27,33)(28,37)(29,39)(30,31)(32,36)(34,40)(35,38);(17,19,21,24)(18,22,20,23)(26,31)(29,37)(32,33)(35,40);(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,23)(21,24)(25,27,30,36)(26,32,31,33)(28,38,39,34)(29,35,37,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,15,16,14)(17,18,21,20)(19,22,24,23)(25,28)(26,29)(27,34)(30,39)(31,37)(32,40)(33,35)(36,38);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(17,20,21,18)(19,23,24,22)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=g,d^2=fgh,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, 1156622);