#6167878

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

1612 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)(9,10,12,14)(11,13,16,15)(17,18,20,22)(19,21,23,24)(26,28)(27,33)(29,37)(39,40);(3,8)(5,7)(9,10)(11,13)(12,14)(15,16)(19,23)(21,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,38,39);(9,10,12,14)(11,15,16,13)(17,19,20,23)(18,21,22,24)(25,27)(26,31)(28,34)(29,38)(30,33)(32,40)(35,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,29,30,37)(26,32,28,36)(27,35,33,38)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=h,d^2=g,e^2=g,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, 6075074);