Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,16,21,15,36)(11,17,25,40,23,38,26,54)(12,27,30,56,32,57,31,60)(13,33,34,44,35,61,28,19)(18,41,42,59,43,58,55,29)(22,45,49,68,47,66,50,71)(24,52,53,65,51,72,46,39)(37,62,64,70,63,69,67,48);(2,5)(6,8)(9,11,14,25,16,23,15,26)(10,17,20,40,21,38,36,54)(12,22,30,49,32,47,31,50)(13,24,34,53,35,51,28,46)(18,37,42,64,43,63,55,67)(19,39,33,52,44,65,61,72)(27,45,56,68,57,66,60,71)(29,48,41,62,59,70,58,69);(9,12,28,58,15,31,35,59,16,32,34,41,14,30,13,29)(10,18,33,56,20,42,44,57,21,43,61,60,36,55,19,27)(11,22,46,69,26,50,51,70,23,47,53,62,25,49,24,48)(17,37,52,68,40,64,65,66,38,63,72,71,54,67,39,45);(1,3,4,7)(2,6,5,8)(11,23)(17,38)(22,47)(24,51)(25,26)(37,63)(39,65)(40,54)(45,66)(46,53)(48,70)(49,50)(52,72)(62,69)(64,67)(68,71)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402511);