#6136603

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6043799);