Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14619 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
686 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)(3,8,15,28)(4,9)(5,10,21,36)(6,23)(7,11)(12,25)(13,38,40,26)(14,27)(16,37,24,29)(17,30,44,55)(18,31,45,58)(19,47)(20,32)(22,50,49,35)(33,48)(34,41)(39,59)(42,60)(43,54)(46,64,61,57)(51,53)(52,62)(56,63);(1,3)(2,8,11,28)(4,17)(5,14,21,41)(6,24)(7,15)(9,30,32,55)(10,34)(12,37,23,29)(13,19)(16,25)(18,43,45,63)(20,44)(22,51,49,42)(26,47,38,33)(27,36)(31,56)(35,53)(39,61,62,46)(40,48)(50,60)(52,64)(54,58)(57,59);(1,4,6,19,7,20,25,48)(2,9,23,47,11,32,12,33)(3,13,16,17,15,40,24,44)(5,18,22,46,21,45,49,61)(8,26,37,55,28,38,29,30)(10,31,50,64,36,58,35,57)(14,39,42,43,41,62,51,63)(27,52,60,56,34,59,53,54);(1,5,6,22,7,21,25,49)(2,10,12,35,11,36,23,50)(3,14,16,42,15,41,24,51)(4,18,19,46,20,45,48,61)(8,27,29,53,28,34,37,60)(9,31,33,57,32,58,47,64)(13,39,17,43,40,62,44,63)(26,52,30,54,38,59,55,56)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=egh,d^2=g,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298371);