#6184939

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

1538 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=fg,d^2=fgh,e^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, 6092135);