#4398780

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1636 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)(17,18,20,22)(19,21,23,24)(25,26)(27,33)(28,30)(29,36)(31,34)(32,38)(35,39)(37,40);(9,11,13,12)(10,14,16,15)(19,23)(21,24)(26,28)(27,34)(32,36)(35,37);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,27)(26,31)(28,33)(29,37)(30,34)(32,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(25,28,30,26)(27,33,34,31)(29,36,38,32)(35,40,37,39);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4305976);