#1784498

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1492 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1691694);