Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2396 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 2396 conjugacy classes of subgroups — too many to draw. Only its 218 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:
(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,18)(19,27)(20,25)(21,28);(2,4)(13,15)(14,19)(16,22)(17,26)(18,27)(20,28)(21,25)(23,24);(7,12)(9,11)(17,24)(20,25)(21,28)(23,26)Polycyclic presentation:
2^9|a^2=e,b^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234968, #72261147, #72263014, … |
| maximal normal subgroup of | #72234968, #72261147, #72263014, … |
| maximal quotient of | #72236996 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325082);