Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152085 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12695 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3509 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
576 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 576 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) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,7,13,25,40)(3,14,16,38,18,31,47,57)(4,19,21,49,20,9,33,35)(6,12,27,39,28,10,24,37)(8,23,30,26,32,46,58,17)(15,42,44,63,43,41,62,60)(22,34,54,59,50,36,52,61)(29,55,56,45,48,51,64,53);(1,3,7,18)(2,8,13,32)(4,15,20,43)(5,23,25,46)(6,26,28,17)(9,29,19,48)(10,31,12,14)(11,38,40,57)(16,24,47,27)(21,51,33,55)(22,53,50,45)(30,39,58,37)(34,42,36,41)(35,60,49,63)(44,52,62,54)(56,61,64,59);(1,4,7,20)(2,9)(3,15,18,43)(5,21,25,33)(6,22,28,50)(8,29)(10,34)(11,35)(12,36)(13,19)(14,41)(16,44,47,62)(17,45,26,53)(23,51,46,55)(24,52,27,54)(30,56)(31,42)(32,48)(37,59)(38,60)(39,61)(40,49)(57,63)(58,64)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59284);