#1310134

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

1452 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=fh,d^2=gi,e^2=h,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, 1217330);