Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#334 |
| prime radical | C23 | #334 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
150 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,12,15,26)(4,9)(5,18)(6,21)(7,11)(8,14,23,28)(10,19)(13,25,16,30)(17,24)(20,31)(22,27,32,29)(33,34,36,40,37,41,46,52,38,42,47,53,48,54,58,62)(35,39,44,50,45,51,57,61,43,49,55,59,56,60,63,64);(1,3,5,14,6,16,19,29,7,15,20,28,4,13,17,27)(2,8,10,12,9,22,18,25,11,23,24,26,21,32,31,30)(33,35)(34,39)(36,44)(37,45)(38,43)(40,50)(41,51)(42,49)(46,57)(47,55)(48,56)(52,61)(53,59)(54,60)(58,63)(62,64);(1,4,7,6)(2,9,11,21)(3,13,15,16)(5,17,20,19)(8,22,23,32)(10,18,24,31)(12,25,26,30)(14,27,28,29)(35,43)(39,49)(44,55)(45,56)(50,59)(51,60)(57,63)(61,64)Polycyclic presentation:
2^9|a^2=e,b^2=d,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ci,e^b=eh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72231941, #72231946, #72231947, … |
| maximal normal subgroup of | #72231941, #72231946, #72231947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30121);