#5493984

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

1440 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)(3,6)(4,5)(7,8)(10,13)(14,16)(19,23)(21,24)(25,26,30,31)(27,32,34,29)(28,33,37,38)(35,39,40,36);(3,7)(6,8)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(26,31)(29,32)(33,38)(36,39);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24)(25,28,30,37)(26,33,31,38)(27,35,34,40)(29,36,32,39);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)(25,29,30,32)(26,34,31,27)(28,36,37,39)(33,40,38,35)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=f,e^2=fg,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, 5401180);