Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #122971 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×C22⋊C8 | C23 |
| prime radical | C25 | D8 |
| LCS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
406 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 44 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)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,31,46)(20,25,22,37)(21,33,36,47)(24,39,42,30)(26,32,44,35)(29,45,38,43)(34,40,48,41);(1,3,5,11)(2,6,8,13)(4,10,12,16)(7,9,14,15)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,40)(26,43)(27,42)(28,33)(30,35)(32,39)(36,48)(37,41)(44,45)(46,47);(4,12)(10,16)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,40,42,41)(26,33,44,47)(28,45,46,43)(30,34,39,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30182);