#4443977

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

1434 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgh,c^2=i,d^2=gi,e^2=g,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, 4351173);