#4437839

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=f,d^2=i,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, 4345035);