#1117533

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

1896 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=fgh,d^2=f,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, 1024729);