Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454369 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,11,15)(7,16,14,10)(8,12,20,18)(9,17,19,13);(5,7,10,6,11,14,16,15)(8,13,19,12,20,17,9,18);(2,4)(5,8,11,20)(6,12,15,18)(7,17,14,13)(9,16,19,10);(5,9,10,8,11,19,16,20)(6,13,14,12,15,17,7,18)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215961, #72219683, #72219691 |
| maximal normal subgroup of | #72215961, #72219683, #72219691 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361567);