Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205751 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#466 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1494 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)(3,5,7,8)(4,6)(9,10)(11,13)(12,18)(14,19)(15,17)(16,21)(20,23)(22,24);(1,3,4,7)(2,5)(6,8)(9,11,13,10,15,19,14,17)(12,18,20,16,22,24,21,23);(2,6)(5,8)(9,12,13,20,15,22,14,21)(10,16,19,24,17,23,11,18);(3,7)(5,8)(9,13,15,14)(10,11,17,19)(12,20,22,21)(16,18,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=hi,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310571, #72310574, #72310579, … |
| maximal normal subgroup of | #72310571, #72310574, #72310579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112956);