Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3710 conjugacy classes of subgroups; only the 546 normal subgroups are shown.
This group has 3710 conjugacy classes of subgroups — too many to draw. Only its 546 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(14,18)(16,23)(19,24)(20,25)(22,27)(26,28);(13,14,16,18,17,20,23,25)(15,19,22,24,21,26,27,28);(2,4)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(6,8)(15,21)(19,26)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287164, #72287172, #72287208, … |
| maximal normal subgroup of | #72287164, #72287172, #72287208, … |
| maximal quotient of | #72287818 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299374);