#2933113

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

1500 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2840309);