Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121143 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1443 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
490 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 106 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)(4,12,14,7)(5,8)(6,11)(10,16,13,15)(17,18)(19,28,31,24)(20,32)(21,36)(22,27)(23,26)(25,34)(29,46,48,42)(30,40,37,41)(33,39,47,45)(35,44)(38,43);(1,3)(2,6,8,9)(4,13)(5,11)(7,15,12,16)(10,14)(17,19,21,30,23,31,22,37)(18,24,36,41,26,28,27,40)(20,33,34,29,35,47,38,48)(25,42,44,39,43,46,32,45);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,20)(18,25,26,43)(19,29)(21,34)(22,38)(23,35)(24,39,28,45)(27,32,36,44)(30,47)(31,48)(33,37)(40,42,41,46)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=gh,f^c=fhi,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28339);