#4439986

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

1598 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=i,e^2=fgh,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, 4347182);