Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
911 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)(7,13,20,38)(8,23,27,41)(9,28)(10,17)(11,33)(12,16)(14,26,40,34)(15,31)(18,45)(19,48,29,59)(21,37,22,43)(24,52)(25,55,57,39)(30,42)(32,44)(35,51)(36,49,63,50)(46,65)(47,62)(53,58,60,56)(54,61)(64,66)(67,68);(5,7,10,22,12,20,11,21)(6,13,33,37,16,38,17,43)(8,24,26,47,27,18,34,51)(9,29,30,49,32,19,31,50)(14,35,41,45,40,62,23,52)(15,36,44,48,42,63,28,59)(25,46,56,67,57,54,58,66)(39,61,53,68,55,65,60,64);(5,8,12,27)(6,14)(7,18)(9,25,32,57)(10,26,11,34)(13,35,38,62)(15,39)(16,40)(17,23)(19,46)(20,24)(21,47)(22,51)(28,60)(29,54)(30,56,31,58)(33,41)(36,61,63,65)(37,45,43,52)(42,55)(44,53)(48,68,59,64)(49,66)(50,67);(2,4)(5,9,12,32)(6,15,16,42)(7,19,20,29)(8,25,27,57)(10,31,11,30)(13,36,38,63)(14,39,40,55)(17,44,33,28)(18,46,24,54)(21,50,22,49)(23,53,41,60)(26,58,34,56)(35,61,62,65)(37,59,43,48)(45,64,52,68)(47,67,51,66)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ei,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340214);