Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #122544 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| prime radical | #1999 | C4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
352 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18,21,26,23,28,22,27)(19,29,32,42,30,45,33,46)(20,34,36,25,38,43,37,44)(24,35,40,31,39,48,41,47);(2,6)(3,9)(10,15)(14,16)(17,19)(18,24)(20,35)(21,33)(22,32)(23,30)(25,42)(26,41)(27,40)(28,39)(29,43)(31,37)(34,45)(36,47)(38,48)(44,46);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(19,30)(24,39)(29,45)(31,47)(32,33)(35,48)(40,41)(42,46)Polycyclic presentation:
2^9|a^2=e,c^2=hi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29788);