#4343871

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2744 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)(25,26,30,31)(27,32,34,39)(28,29,38,33)(35,40,37,36);(10,14)(15,20)(16,21)(22,24)(26,31)(27,34)(29,33)(35,37);(5,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,32)(28,37)(29,36)(30,34)(31,39)(33,40)(35,38);(1,2,3,4)(5,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28)(26,33)(27,35)(29,31)(30,38)(32,36)(34,37)(39,40);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,33)(26,28,31,38)(27,36,34,40)(32,37,39,35)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,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, 4251067);