#2760360

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
automorphism	Aut(G) order 1048576
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

1454 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)(11,15)(14,16)(18,21)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(17,18)(19,22)(20,21)(23,24)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(2,5)(6,7)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19,20,24)(18,22,21,23)(25,27,29,35)(26,30,32,33)(28,36,37,34)(31,39,40,38);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,18)(19,23)(20,21)(22,24)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=hi,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

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, 2667556);