#2073869

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

1496 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=fhi,d^2=fghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1981065);