#4343881

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2603 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,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, 4251077);