#5782872

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

1689 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5690068);