Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7786 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | #31075 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(1,2,7,13,8,12,6,31)(3,14,17,45,19,46,18,39)(4,20,23,42,24,10,22,38)(5,25,30,41,28,43,29,11)(9,32,35,16,36,26,34,53)(15,47,49,63,51,60,50,44)(21,56,27,37,57,40,58,55)(33,52,62,59,54,64,61,48);(1,3)(2,9)(4,15)(5,26)(6,17)(7,18)(8,19)(10,33)(11,39)(12,36)(13,34)(14,43)(16,30)(20,54)(21,52)(22,49)(23,50)(24,51)(25,46)(27,48)(28,32)(29,53)(31,35)(37,63)(38,61)(40,47)(41,45)(42,62)(44,55)(56,60)(57,64)(58,59);(1,4,8,24)(2,10,12,20)(3,15,19,51)(5,27,28,58)(6,23,7,22)(9,33,36,54)(11,40,41,56)(13,42,31,38)(14,44,46,63)(16,52,53,64)(17,50,18,49)(21,29,57,30)(25,37,43,55)(26,48,32,59)(34,62,35,61)(39,47,45,60)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,d^b=dfi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33492);