Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,17,21,18,16,27,20)(15,22,26,24,23,28,25,19);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(15,23)(19,24)(22,28)(25,26);(2,4)(5,8,9,12)(6,10,7,11)(13,15,17,26,18,23,27,25)(14,19,21,22,16,24,20,28);(6,7)(10,11)(13,16)(14,18)(15,24)(17,20)(19,23)(21,27)(22,25)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295639);