Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2288 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)(6,8)(9,10)(11,12)(13,14,17,21,18,22,16,20)(15,23,24,19,26,27,25,28);(1,2,3,4)(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,23)(22,28);(5,6,7,8)(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,27,28,23);(10,12)(14,21)(16,17)(19,23)(20,22)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^d=ei,g^d=gi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264168, #72264211, #72264228, … |
| maximal normal subgroup of | #72264168, #72264211, #72264228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300280);