#6253942

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

1510 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)(3,5)(4,6)(7,8)(10,13)(14,16)(17,18,20,23)(19,21,24,22)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,40,38,39);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,20,24)(18,21,23,22)(26,28)(29,37)(31,33)(34,38);(2,6)(5,8)(9,10,12,13)(11,14,15,16)(17,19,20,24)(18,22,23,21)(25,27,30,35)(26,31,28,33)(29,38,37,34)(32,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgh,c^2=fi,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, 6161138);