Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393963 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,7)(9,10)(11,19)(12,17)(13,18)(14,16)(15,21)(20,24)(22,23);(5,6,8,7)(9,11,13,21)(10,15,18,19)(12,20,14,22)(16,24,17,23);(10,16)(12,14)(15,23)(17,18)(19,24)(20,22);(2,4)(5,7,8,6)(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|b^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261564, #72261868, #72261904 |
| maximal normal subgroup of | #72261564, #72261868, #72261904 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302295);