Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143154 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #23810 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1746 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,23,29)(19,30,34,46)(20,35,38,27)(21,25,36,26)(22,40,37,28)(24,39,45,31)(32,47,48,44)(33,42,41,43);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19)(18,24)(20,32)(21,39)(22,41)(23,34)(25,42)(26,43)(27,46)(28,47)(29,45)(30,35)(31,36)(33,37)(38,48)(40,44);(6,15)(9,16)(18,25)(20,36)(21,38)(22,37)(24,42)(26,29)(27,40)(28,35)(30,47)(31,32)(33,41)(39,48)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72237266, #72308616, #72308622 |
| maximal normal subgroup of | #72237266, #72308616, #72308622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50354);