#5589861

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1618 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)(3,5,7,8)(11,15)(14,16)(17,18,20,21)(19,22,24,23)(25,26)(27,32)(28,34)(29,31)(30,33)(35,39)(36,40)(37,38);(10,13)(14,16)(18,21)(22,23)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,2)(3,5)(4,6)(7,8)(17,19)(18,22)(20,24)(21,23)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fg,d^2=fg,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, 5497057);