#4453492

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3558 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4360688);