#6190473

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1487 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=gh,c^2=fgh,d^2=gh,e^2=fgh,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, 6097669);