Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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 | |
1897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,18,12,17,10,16)(7,19,23,13,24,38,22,37)(8,25,27,44,28,14,34,40)(9,29,31,15,32,42,33,43)(20,47,49,35,50,45,54,55)(21,51,30,46,52,36,53,41)(26,57,60,39,61,62,59,56)(48,63,66,65,67,68,58,64);(1,2)(3,4)(5,7,12,24)(6,13,17,37)(8,20,28,50)(9,30,32,53)(10,23,11,22)(14,35,25,55)(15,41,43,46)(16,38,18,19)(21,33,52,31)(26,58,61,66)(27,54,34,49)(29,51,42,36)(39,64,56,65)(40,47,44,45)(48,60,67,59)(57,63,62,68);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,34)(12,28)(13,35)(15,39)(16,40)(17,25)(18,44)(19,45)(21,48)(22,49)(23,54)(24,50)(29,62)(30,58)(31,59)(32,61)(33,60)(36,63)(37,55)(38,47)(41,64)(42,57)(43,56)(46,65)(51,68)(52,67)(53,66);(5,9,12,32)(6,15,17,43)(7,21,24,52)(8,26,28,61)(10,33,11,31)(13,36,37,51)(14,39,25,56)(16,29,18,42)(19,46,38,41)(20,48,50,67)(22,53,23,30)(27,60,34,59)(35,63,55,68)(40,62,44,57)(45,65,47,64)(49,66,54,58)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290323);