Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,7,9,8,10,11,12)(13,14)(15,21)(16,25)(17,19)(18,23)(20,26)(22,27)(24,28);(13,15)(14,18)(16,22)(17,24)(19,27)(20,28)(21,25)(23,26);(1,2,3,4)(14,19)(18,27)(20,25)(21,28);(15,22)(16,26)(18,28)(20,25)(21,27)(23,24)Polycyclic presentation:
2^9|a^2=g,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,e^d=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72318666, #72318679, #72318702, … |
| maximal normal subgroup of | #72318666, #72318679, #72318702, … |
| maximal quotient of | #72322422 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132628);