Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129627 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29)(20,33,21,26)(22,27)(23,28)(24,30,41,35)(25,34,42,36)(31,45)(32,43)(37,39)(38,44)(40,46,48,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,30)(21,35)(22,37)(23,32)(25,39)(26,43)(27,40)(28,41)(29,33)(31,38)(34,47)(36,46)(42,45)(44,48);(3,9)(6,12)(10,15)(13,16)(17,20,23,21)(18,25,28,42)(19,30,32,35)(22,36,38,34)(24,39,41,45)(26,44,33,27)(29,40,43,48)(31,47,37,46)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,e^b=eg,f^b=fgi,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36825);