#5557669

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1681 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=f,d^2=f,e^2=fhi,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, 5464865);