#1358005

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

1506 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fh,d^2=gh,e^2=gi,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, 1265201);