Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2215 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)(5,6,9,11)(7,12,8,10)(13,14)(15,21)(16,19)(17,20)(18,24)(22,23)(25,27)(26,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,24)(17,26)(19,21)(20,22)(23,25)(27,28);(1,2,3,4)(13,14)(15,22)(16,19)(17,27)(18,26)(20,25)(21,23)(24,28);(5,7,9,8)(6,10,11,12)(14,19)(18,21)(20,27)(22,28);(5,8,9,7)(6,10,11,12)(15,23)(17,25)(18,28)(20,27)(21,22)(24,26)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283038, #72283133, #72283676, … |
| maximal normal subgroup of | #72283038, #72283133, #72283676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593949);