#3630667

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1746 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fg,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3537863);