Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #93290 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1274 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | #1726 | C22 |
| socle | C2 | #7018 |
| prime radical | C23 | #344 |
| LCS | G ▷ C42 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ #1726 ▷ C22×C4 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 162 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 — 2 generators on 64 points:
(1,2)(3,14,16,37,19,40,18,39)(4,20,21,33)(5,10,24,34)(6,11)(7,13)(8,15,27,41,30,43,29,45)(9,23,31,51)(12,25)(17,38,47,61,48,62,46,60)(22,49)(26,35,44,36)(28,54,55,63,56,42,53,64)(32,58)(50,59)(52,57);(1,3,5,17,25,16,44,46,7,19,24,48,6,18,26,47)(2,8,10,28,12,29,36,55,13,30,34,56,11,27,35,53)(4,15,22,42,51,45,58,64,21,43,50,54,23,41,52,63)(9,14,32,38,33,37,59,60,31,40,57,62,20,39,49,61)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=fi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,d^b=dg,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239205, #72239206, #72239207 |
| maximal normal subgroup of | #72239205, #72239206, #72239207 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 503);