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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,22)(16,18)(17,26)(19,21)(20,25)(23,27)(24,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,22)(17,24)(18,25)(19,23)(21,28)(26,27);(2,4)(13,16,18,14)(15,20,25,22)(17,21,19,26)(23,28,24,27);(2,4)(5,8,9,7)(6,10,11,12)(14,16)(15,23)(17,19)(20,27)(22,28)(24,25);(6,11)(10,12)(13,17,18,19)(14,21,16,26)(15,24,25,23)(20,28,22,27)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=fi,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,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282009, #72282899 |
| maximal normal subgroup of | #72282009, #72282899 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424639);