#4416343

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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

1518 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,21)(19,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(2,3)(6,7)(9,10,12,14)(11,13,16,15)(27,33)(31,34)(35,39)(38,40);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(17,19)(18,22)(20,23)(21,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,4,5,8)(2,6,3,7)(9,11,12,16)(10,13,14,15)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4323539);