Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417692 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3216 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 3216 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,16)(11,18)(13,14)(17,24)(19,20)(21,23);(1,2)(3,4)(9,10)(11,15)(12,20)(13,18)(14,16)(17,21)(19,23)(22,24);(2,4)(9,11,15,10)(12,19,22,17)(13,18,14,16)(20,21,24,23);(6,8)(9,12)(10,17)(11,19)(13,23)(14,21)(15,22)(16,24)(18,20)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72267844 |
| maximal normal subgroup of | #72267844 |
| maximal quotient of | #72267846, #72267853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356669);