#4860528

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1394 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4767724);