#4368885

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1920 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)(9,10,13,16)(11,14,12,15)(19,23)(22,24)(25,26)(27,34)(28,32)(29,37)(30,31)(33,38)(35,36)(39,40);(9,11,13,12)(10,14,16,15)(18,21)(22,24)(26,31)(27,32)(29,38)(36,40);(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,19,20,23)(18,22,21,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4276081);