Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,8)(9,11)(10,12)(13,14,18,21,17,20,27,16)(15,22,25,26,23,19,24,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,26)(17,23)(18,25)(20,22)(21,28)(24,27);(1,2,4,3)(5,8,9,10)(6,7,11,12)(13,16,27,20,17,21,18,14)(15,19,25,28,23,22,24,26);(1,3,4,2)(6,11)(7,12)(14,20)(15,23)(16,21)(24,25)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293903);