Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #459 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6,8,11)(3,9,10,12)(17,18)(19,29)(20,32,34,40)(21,35)(22,24)(23,28)(25,30,43,48)(26,37)(27,31)(33,45,36,46)(38,44,47,42)(39,41);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,21,22,23,31,37,39)(18,24,26,29,28,41,35,27)(20,30,36,38,34,48,33,47)(25,40,44,46,43,32,42,45);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20,23,34)(18,25,28,43)(19,30,31,48)(21,36,37,33)(22,38,39,47)(24,40,41,32)(26,44,35,42)(27,45,29,46)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=gi,e^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287685, #72287705, #72287710 |
| maximal normal subgroup of | #72287685, #72287705, #72287710 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44694);