Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,11);(2,4)(9,11,12,10)(13,14,18,21)(15,22,25,28)(16,20,17,26)(19,23,27,24);(6,8)(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(2,4)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264407, #72264450, #72264505, … |
| maximal normal subgroup of | #72264407, #72264450, #72264505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275208);