Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,20)(12,23)(13,16)(14,18)(15,19)(17,21)(22,24);(2,5)(6,8)(10,16,19,18)(17,20,23,24);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,14)(13,21)(15,22)(16,24)(18,20)(19,23);(3,7)(6,8)(9,12,14,11,15,21,13,22)(10,17,16,20,19,23,18,24)Polycyclic presentation:
2^9|a^2=ef,b^2=g,c^2=ef,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339677, #72339683, #72339733 |
| maximal normal subgroup of | #72339677, #72339683, #72339733 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374169);