Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 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)(7,18,23,37,22,44,21,13)(8,24,27,39,28,14,34,40)(9,29,31,41,32,60,30,15)(19,42,46,35,25,54,51,52)(20,47,50,64,49,36,48,43)(26,56,57,53,58,59,61,38)(45,65,66,63,55,62,67,68);(1,2,3,4)(5,7)(6,13,17,37)(8,25,28,19)(9,20)(10,21)(11,23)(12,22)(14,35)(15,36,41,47)(16,18,33,44)(24,52)(26,55,58,45)(27,51,34,46)(29,43,60,64)(30,48)(31,50)(32,49)(38,62)(39,42)(40,54)(53,65)(56,68)(57,67,61,66)(59,63);(2,4)(5,8)(6,14)(7,19)(9,26)(10,27)(11,34)(12,28)(13,35)(15,38)(16,39)(17,24)(18,42)(20,45)(21,46)(22,25)(23,51)(29,59)(30,57)(31,61)(32,58)(33,40)(36,62)(37,52)(41,53)(43,63)(44,54)(47,65)(48,66)(49,55)(50,67)(56,60)(64,68);(5,9,11,31,12,32,10,30)(6,15,33,60,17,41,16,29)(7,20,23,50,22,49,21,48)(8,26,34,61,28,58,27,57)(13,36,44,64,37,47,18,43)(14,38,40,56,24,53,39,59)(19,45,51,67,25,55,46,66)(35,62,54,68,52,65,42,63)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274287);