#5486271

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=fg,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=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, 5393467);