#4448523

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

1486 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)(9,10,14,13)(11,16,12,15)(17,18,20,22)(19,21,23,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(1,2,5,4)(3,6,7,8)(19,23)(21,24)(26,30)(31,33)(32,36)(39,40);(9,11,14,12)(10,15,13,16)(17,19)(18,21)(20,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,13,15)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,16,12,15)(17,18)(19,21)(20,22)(23,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=g,d^2=fg,e^2=i,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, 4355719);