Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(2,3)(5,7,10,6)(8,12,9,11);(1,2,4,3)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(5,8,10,9)(6,11,7,12)(13,15,17,22)(14,19,21,26)(16,23,24,28)(18,25,20,27);(1,3,4,2)(5,9,10,8)(6,11,7,12)(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718829);