Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| 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 | |
1365 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)(8,23)(9,29)(10,16)(11,17)(12,21)(13,27)(14,32)(15,35)(19,48)(20,44)(22,38)(24,55)(25,54)(26,43)(28,42)(30,37)(31,45)(33,39)(34,46)(36,51)(40,50)(41,57)(47,52)(49,64)(53,62)(56,66)(58,60)(59,65)(61,68)(63,67);(5,7,11,22)(6,12,17,39)(8,24,28,50)(9,30,32,53)(10,33,35,21)(13,40,43,55)(14,44,29,47)(15,38,16,18)(19,27,51,26)(20,34,52,31)(23,36,42,48)(25,49,59,68)(37,46,62,45)(41,64,60,61)(54,63,65,66)(56,58,67,57);(5,8,11,28)(6,13)(7,19,22,51)(9,25,32,59)(10,27,35,26)(12,36)(14,41)(15,23)(16,42)(17,43)(18,40)(20,49,52,68)(21,50,33,24)(29,60)(30,56,53,67)(31,58,34,57)(37,61)(38,55)(39,48)(44,66)(45,65)(46,54)(47,63)(62,64);(5,9,11,32)(6,14,17,29)(7,20,22,52)(8,25,28,59)(10,34,35,31)(12,37,39,62)(13,41,43,60)(15,46,16,45)(18,47,38,44)(19,49,51,68)(21,53,33,30)(23,54,42,65)(24,56,50,67)(26,58,27,57)(36,61,48,64)(40,63,55,66)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293126);