Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2815 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1468 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1468 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,8,11,12)(9,10)(13,14)(15,23,25,27)(16,21)(17,20)(18,22)(19,24,28,26);(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,23)(21,27)(22,28);(5,8)(6,7)(9,12)(10,11)(14,20)(16,17)(19,27)(21,22)(23,28)(24,26)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238062, #72267758, #72267777 |
| maximal normal subgroup of | #72238062, #72267758, #72267777 |
| maximal quotient of | #72238063, #72238070 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18518);