Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #494749 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,11,6,13)(3,15,20,46,21,44,19,45)(4,22,25,10,26,39,24,37)(5,27,16,42,30,12,31,33)(9,32,36,28,35,54,41,18)(17,48,51,43,52,63,50,61)(23,57,47,38,29,56,58,40)(34,53,62,64,60,49,55,59);(1,3,16,32)(2,9,33,46)(4,17,47,59)(5,28,7,19)(6,20,30,18)(8,21,31,54)(10,34,56,48)(11,35,42,45)(12,15,14,36)(13,41,27,44)(22,55,40,43)(23,53,25,50)(24,51,29,49)(26,52,58,64)(37,60,57,63)(38,61,39,62);(1,4,8,26)(2,10,11,37)(3,17,21,52)(5,29,30,23)(6,25,7,24)(9,34,35,60)(12,40,27,38)(13,39,14,22)(15,43,44,61)(16,47,31,58)(18,53,28,49)(19,51,20,50)(32,59,54,64)(33,56,42,57)(36,55,41,62)(45,63,46,48);(2,11)(9,35)(10,37)(12,27)(13,14)(15,44)(22,39)(33,42)(34,60)(36,41)(38,40)(43,61)(45,46)(48,63)(55,62)(56,57)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72280647, #72281917, #72282072, … |
| maximal normal subgroup of | #72280647, #72281917, #72282072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401955);