#4445880

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

1584 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fh,d^2=fg,e^2=fi,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, 4353076);