Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(1,2,3,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(15,21)(18,24)(22,27)(25,28);(14,19)(18,24)(20,26)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,g^2=i,b^a=bf,e^a=ei,d^c=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72239210, #72239412, #72239663, … |
| maximal normal subgroup of | #72239210, #72239412, #72239663, … |
| maximal quotient of | #72237206 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276972);