#1277448

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

1614 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)(11,15)(12,14)(13,16)(17,18,22,19)(20,24,21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(9,11,14,13)(10,15,12,16)(17,19,22,18)(20,24,21,23)(26,31)(27,33)(34,39)(35,40);(1,2,5,4)(3,6,7,8)(9,12,14,10)(11,15,13,16)(17,20,22,21)(18,23,19,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,3)(2,6)(4,8)(5,7)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,21,22,20)(18,24,19,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=fghi,b^2=gi,c^2=fg,d^2=f,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, 1184644);