#6135653

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

1966 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=f,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, 6042849);