Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| 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 | |
1592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,18)(8,23)(9,29)(10,33)(11,16)(12,17)(13,21)(14,27)(15,30)(19,35)(20,48)(22,37)(24,39)(25,40)(26,42)(28,41)(31,43)(32,57)(34,38)(36,50)(44,49)(45,63)(46,60)(47,61)(51,53)(52,67)(54,56)(55,64)(58,62)(59,65)(66,68);(1,3,4,2)(5,7,10,21,12,22,11,34)(6,13,16,18,17,38,33,37)(8,24,26,19,28,46,27,47)(9,20,30,49,32,50,31,58)(14,39,41,61,42,60,23,35)(15,36,29,44,43,48,57,62)(25,52,53,45,55,65,54,66)(40,63,56,67,64,68,51,59);(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,28)(13,35)(15,40)(16,41)(17,42)(18,39)(20,45)(21,46)(22,47)(23,33)(24,34)(29,56)(30,53)(31,54)(32,55)(36,59)(37,60)(38,61)(43,64)(44,63)(48,67)(49,65)(50,66)(51,57)(52,58)(62,68);(5,9,11,31,12,32,10,30)(6,15,33,57,17,43,16,29)(7,20,34,58,22,50,21,49)(8,25,27,54,28,55,26,53)(13,36,37,62,38,48,18,44)(14,40,23,51,42,64,41,56)(19,45,24,52,47,66,46,65)(35,59,60,68,61,67,39,63)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272260);