#1324487

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1566 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fi,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,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, 1231683);