Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128065 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8101 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,11)(3,13,16,43)(4,18,20,39)(5,22,24,40)(7,12,28,41)(8,29,31,49)(9,26,34,21)(10,27,37,25)(14,42,45,63)(15,32,48,38)(17,35,51,60)(19,33,53,61)(23,36,54,62)(30,56,58,44)(46,57,50,64)(47,52,55,59);(1,3)(2,8)(4,14)(5,15)(6,16,28,51)(7,17)(9,30)(10,35)(11,38,41,31)(12,32)(13,22)(18,52)(19,44)(20,45,26,50)(21,46)(23,47)(24,48,27,29)(25,49)(33,42)(34,59,39,58)(36,57)(37,43,40,60)(53,56,54,55)(61,64,62,63);(1,4,7,21)(2,9,12,18)(3,14,17,46)(5,23,25,19)(6,26,28,20)(8,30,32,52)(10,36,22,33)(11,39,41,34)(13,42,35,57)(15,47,49,44)(16,50,51,45)(24,53,27,54)(29,55,48,56)(31,59,38,58)(37,61,40,62)(43,64,60,63)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,d^c=di,e^c=ei,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72303150, #72303154, #72303158, … |
| maximal normal subgroup of | #72303150, #72303154, #72303158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35276);