#4397036

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1842 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=hi,d^2=gi,e^2=fg,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, 4304232);