Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#466 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,9,15,12,17,11,35)(7,18,22,49,23,45,20,48)(8,24,27,44,28,42,25,14)(10,30,29,37,34,61,33,16)(13,36,40,21,41,56,39,31)(19,50,51,64,54,67,53,47)(26,43,60,58,32,46,59,66)(38,52,63,68,65,55,57,62);(5,7,10,31,11,22,33,36,12,23,34,21,9,20,29,56)(6,13,37,18,15,39,61,48,17,41,16,45,35,40,30,49)(8,19,32,52,27,53,59,62,28,54,26,55,25,51,60,68)(14,38,43,47,42,63,66,50,44,65,46,64,24,57,58,67);(2,4)(5,8)(6,14)(7,19)(9,25)(10,32)(11,27)(12,28)(13,38)(15,42)(16,46)(17,44)(18,47)(20,51)(21,55)(22,53)(23,54)(24,35)(26,34)(29,60)(30,58)(31,52)(33,59)(36,62)(37,43)(39,63)(40,57)(41,65)(45,64)(48,50)(49,67)(56,68)(61,66);(5,9,12,11)(6,15,17,35)(7,20,23,22)(8,25,28,27)(10,29,34,33)(13,39,41,40)(14,42,44,24)(16,30,37,61)(18,48,45,49)(19,51,54,53)(21,36,31,56)(26,59,32,60)(38,63,65,57)(43,66,46,58)(47,50,64,67)(52,68,55,62)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401854);