#3265798

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

1748 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)(3,7)(4,6)(5,8)(11,15)(14,16)(18,21)(19,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,3,6,5)(2,7,4,8)(19,23)(22,24)(26,28)(29,37)(31,33)(34,40);(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,15,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)(25,28,30,26)(27,31,35,33)(29,32,37,36)(34,39,40,38);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,16,15,14)(17,18)(19,22)(20,21)(23,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=ghi,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3172994);