Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| 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 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14)(15,23)(16,21)(17,19)(18,22)(20,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(14,19)(15,16)(18,25)(20,21)(23,27)(24,26);(1,2,3,4)(6,9)(10,12)(13,15,18,16,17,24,25,26)(14,20,22,27,19,23,28,21);(5,7,8,11)(6,10,9,12)(14,19)(20,23)(21,27)(22,28);(13,16,25,15,17,26,18,24)(14,21,28,23,19,27,22,20)Polycyclic presentation:
2^9|c^2=fgh,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308331, #72308353 |
| maximal normal subgroup of | #72308331, #72308353 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724241);