#4450702

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

1722 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=f,d^2=fgh,e^2=fgi,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, 4357898);