#4615081

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

1562 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)(3,7,8,5)(11,15)(14,16)(19,23)(21,24)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,40,37,36);(1,2,4,6)(3,5,8,7)(10,13)(14,16)(17,18,20,22)(19,21,23,24)(26,31)(27,34)(33,39)(35,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(17,19)(18,21)(20,23)(22,24)(25,28,30,32)(26,27,31,34)(29,36,38,40)(33,35,39,37);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=f,e^2=g,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, 4522277);