#5743729

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1679 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5650925);