#4427702

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

1488 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)(9,10)(11,14)(12,13)(15,16)(18,22)(23,24)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,2,6,5)(3,7,4,8)(9,10,13,12)(11,15,16,14)(27,34)(31,33)(35,40)(38,39);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,22)(19,23,20,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,4,6,3)(2,8,5,7)(9,12,13,10)(11,15,16,14)(17,19,21,20)(18,23,22,24)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(1,5,6,2)(3,8,4,7)(9,11)(10,15)(12,14)(13,16)(17,20,21,19)(18,24,22,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=g,d^2=fg,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, 4334898);