Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3058 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #24267 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
946 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 946 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 64 points:
(1,2)(3,13)(4,19)(5,23)(6,26)(7,11)(8,16)(9,21)(10,14)(12,24)(15,40)(17,33)(18,46)(20,47)(22,52)(25,53)(27,45)(28,55)(29,57)(30,50)(31,58)(32,39)(34,42)(35,37)(36,44)(38,48)(41,56)(43,54)(49,51)(59,63)(60,64)(61,62);(1,3,14,37,6,18,25,45,7,17,39,57,24,44,5,16)(2,8,23,36,12,29,32,33,11,27,53,46,26,35,10,13)(4,15,38,61,22,43,50,63,20,41,58,64,51,55,21,42)(9,28,49,60,31,56,47,59,30,54,52,62,48,40,19,34);(4,20)(8,27)(9,30)(13,33)(15,41)(19,47)(21,50)(22,51)(29,35)(31,48)(36,46)(38,58)(42,63)(43,55)(49,52)(61,64);(1,4,14,38,6,22,25,50,7,20,39,58,24,51,5,21)(2,9,23,49,12,31,32,47,11,30,53,52,26,48,10,19)(3,15,37,61,18,43,45,63,17,41,57,64,44,55,16,42)(8,28,36,60,29,56,33,59,27,54,46,62,35,40,13,34)Polycyclic presentation:
2^9|b^2=eh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72217290, #72224994, #72225014, … |
| maximal normal subgroup of | #72217290, #72224994, #72225014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408047);