Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121129 |
| 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 | |
500 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 108 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)(20,25)(21,36)(22,27)(23,26)(24,30)(29,44)(31,45)(32,39)(33,38)(34,42)(35,43)(37,41)(40,46)(47,48);(1,3)(2,6,8,9)(4,13)(5,11)(7,15,12,16)(10,14)(17,19,23,32,22,31,21,30)(18,24,36,45,27,39,26,28)(20,33,34,46,35,47,37,29)(25,40,41,38,43,44,42,48);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,20,21,34,22,35,23,37)(18,25,26,41,27,43,36,42)(19,29,30,33,31,46,32,47)(24,38,28,44,39,48,45,40)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,e^c=eh,f^c=fhi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28325);