#5486456

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1890 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)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)(26,30)(27,33)(28,36)(39,40);(3,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,21,22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(2,5)(6,8)(10,15)(11,16)(18,22)(21,24)(25,27,29,33)(26,31,30,34)(28,37,36,35)(32,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,21)(20,23)(22,24)(25,28,29,36)(26,32,30,38)(27,35,33,37)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=gi,c^2=f,e^2=f,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, 5393652);