Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,33)(7,18,22,37,23,46,21,13)(8,24,26,39,27,14,34,40)(9,28,31,15,32,42,30,43)(19,47,49,55,50,44,54,35)(20,51,52,45,53,36,29,41)(25,57,59,38,60,62,61,56)(48,63,58,65,66,68,67,64);(1,2)(3,4)(5,7,10,21,12,23,11,22)(6,13,16,18,17,37,33,46)(8,19,26,49,27,50,34,54)(9,29,30,53,32,52,31,20)(14,35,39,44,24,55,40,47)(15,41,42,51,43,45,28,36)(25,58,59,66,60,67,61,48)(38,64,57,68,56,65,62,63);(2,4)(5,8)(6,14)(7,19)(9,25)(10,26)(11,34)(12,27)(13,35)(15,38)(16,39)(17,24)(18,44)(20,48)(21,49)(22,54)(23,50)(28,62)(29,58)(30,59)(31,61)(32,60)(33,40)(36,63)(37,55)(41,64)(42,57)(43,56)(45,65)(46,47)(51,68)(52,67)(53,66);(5,9,12,32)(6,15,17,43)(7,20,23,53)(8,25,27,60)(10,31,11,30)(13,36,37,51)(14,38,24,56)(16,28,33,42)(18,45,46,41)(19,48,50,66)(21,52,22,29)(26,61,34,59)(35,63,55,68)(39,62,40,57)(44,65,47,64)(49,67,54,58)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^c=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, 290415);