Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,23,27,25)(16,21,18,17)(20,24,28,26);(2,4)(13,15,16,24,19,27,18,26)(14,20,17,25,22,28,21,23);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,22,17)(15,24,27,26)(20,23,28,25);(6,9)(10,12)(13,17,18,14,19,21,16,22)(15,25,26,20,27,23,24,28)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715345);