Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110189 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1437 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
974 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 214 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)(3,8)(4,5)(9,10)(11,17,20,15)(12,21)(13,22)(14,16)(18,19,24,23);(1,2,5,3,6,8,4,7)(9,11)(10,15)(12,18)(13,23)(14,20)(16,17)(19,21)(22,24);(1,3,4,2,6,7,5,8)(9,12,21,10,14,22,13,16)(11,18,19,15,20,24,23,17)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,e^b=ei,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214129, #72214439, #72214488, … |
| maximal normal subgroup of | #72214129, #72214439, #72214488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17412);