Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129437 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1218 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,14)(8,12,17,10)(9,13)(11,16)(15,18,20,19);(1,2)(3,4)(6,10)(8,17)(11,18)(12,13)(14,19)(15,20);(2,4)(5,7,9,16)(6,11)(8,15,17,20)(10,18)(12,19)(13,14)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,g^b=ghi,e^c=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72249497, #72249504, #72249506, … |
| maximal normal subgroup of | #72249497, #72249504, #72249506, … |
| maximal quotient of | #72255789, #72261953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54801);