Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2343 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)(5,6)(7,10)(8,12)(9,11);(5,7,10,6)(8,12,9,11)(14,18)(15,21)(16,24)(19,22)(20,26)(23,27);(2,4)(13,14,16,18,17,20,24,26)(15,22,21,28,23,19,27,25);(5,8,10,9)(6,11,7,12)(15,23)(19,22)(21,27)(25,28);(5,9,10,8)(6,11,7,12)(13,15,16,21,17,23,24,27)(14,19,18,25,20,22,26,28)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308113, #72308452, #72308923 |
| maximal normal subgroup of | #72308113, #72308452, #72308923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718739);