Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | |
1972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,19)(17,26)(18,21)(20,25)(23,24)(27,28);(2,3)(6,9)(10,12)(13,15)(14,20)(16,23)(17,24)(18,25)(19,27)(21,22)(26,28);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,18,26)(14,19,21,17)(15,23,25,28)(20,27,22,24);(1,3,4,2)(13,14)(15,23)(16,17)(18,21)(19,26)(20,27)(22,24)(25,28);(6,9)(10,12)(13,17,18,19)(14,16,21,26)(15,24,25,27)(20,23,22,28)Polycyclic presentation:
2^9|c^2=fg,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72307384, #72307674, #72308310 |
| maximal normal subgroup of | #72307384, #72307674, #72308310 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467684);