Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511825 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13)(4,18)(5,10)(6,24)(7,12)(8,16)(9,20)(11,26)(14,35)(15,41)(17,45)(19,48)(21,34)(22,32)(23,52)(25,31)(27,39)(28,56)(29,57)(30,50)(33,51)(36,43)(37,40)(38,62)(42,47)(44,54)(46,61)(49,60)(53,59)(55,64)(58,63);(1,3,7,14)(2,8,12,27)(4,15,21,38)(5,16,22,39)(6,17,26,40)(9,28,31,55)(10,13,32,35)(11,29,24,54)(18,47,34,58)(19,42,49,63)(20,43,25,46)(23,44,51,57)(30,36,59,61)(33,37,52,45)(41,60,62,48)(50,56,53,64);(3,14)(5,22)(8,27)(13,35)(15,38)(17,40)(19,49)(23,51)(28,55)(29,54)(36,61)(37,45)(41,62)(43,46)(47,58)(50,53);(1,4,7,21)(2,9,12,31)(3,15,14,38)(5,19,22,49)(6,25,26,20)(8,28,27,55)(10,30,32,59)(11,34,24,18)(13,36,35,61)(16,42,39,63)(17,46,40,43)(23,53,51,50)(29,58,54,47)(33,60,52,48)(37,62,45,41)(44,64,57,56)Polycyclic presentation:
2^9|b^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72260417, C2×#511828, #72303715, … |
| maximal normal subgroup of | #72260417, C2×#511828, #72303715, … |
| maximal quotient of | #72260477, #72261262, C2×#511828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419024);