Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
14970 conjugacy classes of subgroups; only the 3962 normal subgroups are shown.
This group has 14970 conjugacy classes of subgroups — too many to draw. Only its 3962 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 — 6 generators on 24 points:
(2,4)(6,7)(12,20)(16,21)(18,24)(22,23);(5,6,8,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24);(1,2,3,4)(5,7,8,6)(10,14)(15,17)(16,21)(22,23);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72267577, #72267585, C2×#7630463 |
| maximal normal subgroup of | #72267577, #72267585, C2×#7630463 |
| maximal quotient of | #72323283, #72323319, C2×#7630463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537659);