#4430371

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1710 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4337567);