#6136586

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1683 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)(10,12)(14,15)(17,18,22,20)(19,24,21,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,40,38,39);(2,5)(6,8)(10,12)(14,15)(17,18,22,20)(19,23,21,24)(26,28)(29,37)(31,33)(34,38);(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,23,20,24)(25,27,30,35)(26,31,28,33)(29,38,37,34)(32,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,20,22,18)(19,23,21,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,21,22,19)(18,23,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=g,d^2=g,e^2=fg,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, 6043782);