Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
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×H(5,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
17422 conjugacy classes of subgroups; only the 738 normal subgroups are shown.
This group has 17422 conjugacy classes of subgroups — too many to draw. Only its 738 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(6,8)(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72307823, #72307875, #72307909, … |
| maximal normal subgroup of | #72307823, #72307875, #72307909, … |
| maximal quotient of | #72314578 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299317);