#1279824

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fhi,c^2=fg,d^2=fh,e^2=f,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, 1187020);