#5203517

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1734 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=fi,d^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5110713);