Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | #11460 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,12,6,14,26,39,7,11,25,36,27,37,52,59,8,13,24,38,28,40,51,60,4,10,21,34,22,35,48,58)(3,15,17,41,18,42,45,61,16,9,29,30,43,32,53,55,19,33,23,49,46,57,50,63,20,31,44,54,47,56,62,64);(1,3)(2,9)(4,20)(5,23)(6,18)(7,16)(8,19)(10,15)(11,33)(12,30)(13,31)(14,32)(17,24)(21,29)(22,47)(25,44)(26,50)(27,43)(28,46)(34,41)(35,42)(36,49)(37,57)(38,54)(39,55)(40,56)(45,51)(48,53)(52,62)(58,61)(59,63)(60,64);(2,10,13,11)(5,24)(9,15,31,33)(12,36,38,34)(14,35,40,37)(17,23)(21,25)(26,51)(29,44)(30,49,54,41)(32,42,56,57)(39,59,60,58)(45,50)(48,52)(53,62)(55,63,64,61)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,d^b=di,e^b=efi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287335, #72287376, #72287401, … |
| maximal normal subgroup of | #72287335, #72287376, #72287401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56857);