Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #343051 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2679 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #24222 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
815 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 815 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 4 generators on 24 points:
(1,2,3,6,4,7,8,5)(9,10,12,18,14,19,13,15)(11,16,20,23,21,24,22,17);(10,15)(11,20)(12,13)(17,23)(18,19)(21,22);(2,5)(3,8)(6,7)(9,11)(10,16)(12,20)(13,22)(14,21)(15,17)(18,23)(19,24);(2,6)(3,8)(5,7)(9,11)(10,17)(12,22)(13,20)(14,21)(15,16)(18,24)(19,23)Polycyclic presentation:
2^9|a^2=ef,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213461 |
| maximal normal subgroup of | #72213461 |
| maximal quotient of | #72213348 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 250247);