Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#323 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
326 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,11)(4,8)(5,18,19,9)(6,10)(7,13,21,28)(12,25)(14,26)(15,29)(16,31,32,23)(17,24)(20,27,22,30)(33,34,37,43,38,44,52,61,39,41,53,57,36,42,50,59)(35,45,47,62,46,55,63,60,48,58,51,64,49,40,54,56);(1,3,6,14)(2,7,10,21)(4,15,17,12)(5,13,19,28)(8,22,24,20)(9,11,18,26)(16,30,32,27)(23,29,31,25)(33,35)(34,40)(36,49)(37,51)(38,46)(39,48)(41,55)(42,58)(43,60)(44,45)(47,53)(50,63)(52,54)(56,57)(59,62)(61,64);(1,4,6,17)(2,8,10,24)(3,12,14,15)(5,16,19,32)(7,20,21,22)(9,23,18,31)(11,25,26,29)(13,27,28,30)(34,41)(40,55)(42,44)(43,57)(45,58)(56,60)(59,61)(62,64)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=h,d^2=i,e^2=g,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=ch,d^b=di,e^b=efi,e^d=eh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29465);