Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1442 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
492 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7,8,3)(9,10,13,17,14,18,23,24)(11,15,19,21,20,12,16,22);(1,2,5,7,6,8,4,3)(9,11)(10,15)(12,18)(13,19)(14,20)(16,23)(17,21)(22,24);(1,3,4,2,6,7,5,8)(9,12)(10,16)(11,17)(13,22)(14,15)(18,19)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fi,g^b=gh,e^c=eh,f^c=fi,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265410, #72265464, #72265468 |
| maximal normal subgroup of | #72265410, #72265464, #72265468 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27828);