Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1359 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)(7,18)(8,24)(9,30)(10,35)(11,16)(12,17)(13,21)(14,28)(15,32)(19,47)(20,52)(22,44)(23,38)(25,36)(26,57)(27,41)(29,45)(31,42)(33,43)(34,46)(37,53)(39,51)(40,59)(48,54)(49,68)(50,65)(55,61)(56,67)(58,64)(60,62)(63,66);(5,7)(6,13)(8,25)(9,31)(10,23)(11,21)(12,22)(14,39)(15,42)(16,44)(17,38)(18,35)(19,29)(20,32)(24,47)(26,58)(27,51)(28,50)(30,37)(33,53)(34,54)(36,41)(40,64)(43,48)(45,65)(46,52)(49,60)(55,68)(56,59)(57,66)(61,63)(62,67);(5,8,12,28)(6,14)(7,19,22,51)(9,26,34,60)(10,29,11,27)(13,36)(15,40)(16,45)(17,24)(18,47)(20,49,53,58)(21,25,23,50)(30,62)(31,63,54,56)(32,61,33,59)(35,41)(37,64)(38,65)(39,44)(42,66)(43,55)(46,57)(48,67)(52,68);(5,9,12,34)(6,15,17,43)(7,20,22,53)(8,26,28,60)(10,33,11,32)(13,37,38,52)(14,40,24,55)(16,46,35,30)(18,48,44,42)(19,49,51,58)(21,54,23,31)(25,56,50,63)(27,61,29,59)(36,64,65,68)(39,66,47,67)(41,62,45,57)Polycyclic presentation:
2^9|a^2=h,c^2=egi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293118);