Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151506 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12291 |
| commutator, Frattini, cosocle | C2×M32 | C23 |
| prime radical | #2218 | C22 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 42 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:
(1,2,8,13,6,14,7,30)(3,15,31,47,17,10,19,38)(4,20,32,43,24,9,26,35)(5,27,23,40,29,41,28,12)(11,39,34,21,42,57,55,25)(16,33,56,60,22,44,51,62)(18,53,49,37,52,58,54,46)(36,61,59,50,45,64,63,48);(2,9)(4,21)(5,28)(7,8)(10,33)(11,40)(12,42)(13,43)(14,20)(15,44)(16,48)(18,54)(19,31)(22,50)(23,29)(24,25)(26,57)(27,34)(30,35)(32,39)(36,46)(37,45)(38,62)(41,55)(47,60)(49,52)(51,61)(53,63)(56,64)(58,59);(1,3)(2,10,14,15)(4,22)(5,18)(6,17)(7,19)(8,31)(9,33,20,44)(11,36,42,45)(12,37,40,46)(13,38,30,47)(16,24)(21,50)(23,49)(25,48)(26,56)(27,58,41,53)(28,54)(29,52)(32,51)(34,59,55,63)(35,60,43,62)(39,61)(57,64)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=fhi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,f^b=fh,h^b=hi,d^c=dg,e^c=ei,e^d=ei38 conjugacy classes — view the full character table.
| maximal subgroup of | #72320858, #72320907, #72320934, … |
| maximal normal subgroup of | #72320858, #72320907, #72320934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58704);