Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,21)(20,25)(22,28)(23,27)(24,26);(14,18,21,20)(15,22,23,24)(19,27)(26,28);(1,2,4,3)(6,9)(10,12);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,24)(17,23)(18,26)(20,28)(21,27)(22,25)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=fg,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264075);