Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,15,18,21)(12,20,16,22);(1,2)(3,4)(9,10)(11,16)(12,19)(13,21)(14,18)(15,23)(17,20)(22,24);(2,4)(6,8)(9,11,13,17,14,19,23,24)(10,16,21,20,18,12,15,22);(6,8)(9,12)(10,17)(11,15)(13,22)(14,16)(18,24)(19,21)(20,23)Polycyclic presentation:
2^9|a^2=h,c^2=efh,e^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
| maximal quotient of | #72240061, #72240063, #72240064 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296337);