#2227997

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1562 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2135193);