Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133912 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2310 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
543 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 78 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)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18,21,26,22,27,38,45)(19,28,31,46,30,24,36,42)(20,32,34,44,35,25,37,43)(23,39,41,33,40,47,48,29);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,23,27,40)(20,29)(21,36)(22,30)(24,25,28,32)(26,41,45,48)(31,38)(33,35)(34,47)(37,39)(42,43,46,44);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,19)(18,24,27,28)(20,33)(21,31)(22,30)(23,25,40,32)(26,42,45,46)(29,35)(34,47)(36,38)(37,39)(41,43,48,44)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41198);