Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
752 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18,22,28,23,27,21,26)(19,24,32,40,31,41,30,39)(20,33,37,25,35,43,36,42)(29,44,46,38,34,47,45,48);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,34,35,29)(21,30)(22,32)(23,31)(25,38,42,48)(26,39)(27,41)(28,40)(33,44,43,47)(36,46,37,45);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,20)(18,25)(19,29)(21,37)(22,36)(23,35)(24,38)(26,43)(27,42)(28,33)(30,46)(31,34)(32,45)(39,47)(40,44)(41,48)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23560);