#4444433

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

1426 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=gi,d^2=hi,e^2=fh,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, 4351629);