Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1607 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1607 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,23)(22,24);(3,7)(5,8)(10,15)(12,13)(16,19)(17,18)(20,22)(23,24);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,20)(13,22)(14,21)(15,23)(17,19)(18,24);(3,7)(5,8)(9,11)(10,16)(12,22)(13,20)(14,21)(15,19)(17,23)(18,24)Polycyclic presentation:
2^9|e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213362, #72213674, #72213691, … |
| maximal normal subgroup of | #72213362, #72213674, #72213691, … |
| maximal quotient of | #72338276, #72339617 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154183);