#4969543

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1430 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:

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

Polycyclic presentation:

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

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, 4876739);