Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,22,18,21,25,28)(15,19,23,27,24,26,16,20);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,28)(21,26)(22,27);(5,7,9,8)(6,11,12,10)(13,16,17,15,18,23,25,24)(14,20,22,19,21,27,28,26);(2,4)(5,8,9,7)(6,10,12,11)(14,21)(19,26)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=efh,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297399);