Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,16)(15,18,20,19)(21,22)(23,28)(24,30)(25,29)(26,27)(31,35)(32,36)(33,34);(5,6,8,10)(7,11,12,9)(13,14,17,16)(15,19,20,18)(22,27)(23,30)(25,33)(32,35);(5,6,8,10)(7,9,12,11)(13,15,17,20)(14,18,16,19)(21,23,26,30)(22,28,27,24)(25,31,33,36)(29,35,34,32);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,24,26,28)(22,23,27,30)(25,32,33,35)(29,31,34,36);(5,7)(6,9)(8,12)(10,11)(13,16,17,14)(15,19,20,18)(21,25,26,33)(22,29,27,34)(23,31,30,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=gh,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1654266);