Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495197 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,18,41,19,42,29,40)(4,20,24,36,21,50,23,10)(5,25,15,38,27,11,30,32)(9,31,34,26,35,48,58,17)(16,44,47,39,45,62,55,61)(22,53,43,57,52,51,56,37)(33,46,49,59,60,54,64,63);(1,3,15,31)(2,9,32,41)(4,16,43,59)(5,26,7,29)(6,18,27,17)(8,19,30,48)(10,33,53,62)(11,14,12,34)(13,35,38,40)(20,49,37,39)(21,45,56,63)(22,54,24,55)(23,47,52,46)(25,42,28,58)(36,60,51,44)(50,64,57,61);(1,4,6,23,8,21,7,24)(2,10,12,20,13,36,28,50)(3,16,18,47,19,45,29,55)(5,22,15,43,27,52,30,56)(9,33,34,49,35,60,58,64)(11,37,38,51,25,57,32,53)(14,39,40,44,42,61,41,62)(17,46,48,63,26,54,31,59);(4,21)(10,36)(16,45)(20,50)(22,52)(23,24)(33,60)(37,57)(39,61)(43,56)(44,62)(46,54)(47,55)(49,64)(51,53)(59,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72312588, #72312609, #72312645, … |
| maximal normal subgroup of | #72312588, #72312609, #72312645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402399);