Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×D4×D8 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(5,6)(7,8)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(2,4)(6,8);(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^d=eh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257068, C2×#499713, #72310184, … |
| maximal normal subgroup of | #72257068, C2×#499713, #72310184, … |
| maximal quotient of | #72265548, C2×#499713, #72322628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406909);