Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,4,3)(5,6,11,16,12,18,10,17)(7,19,22,13,24,36,23,37)(8,25,28,14,29,39,27,40)(9,15,32,42,33,44,31,43)(20,34,47,52,49,60,48,45)(21,41,50,35,51,61,30,58)(26,53,56,38,57,63,55,64)(46,59,65,67,66,68,54,62);(1,3,4,2)(5,7)(6,13)(8,20)(9,30)(10,22)(11,23)(12,24)(14,34)(15,41)(16,19)(17,36)(18,37)(21,31)(25,52)(26,54)(27,47)(28,48)(29,49)(32,51)(33,50)(35,43)(38,62)(39,45)(40,60)(42,58)(44,61)(46,55)(53,59)(56,66)(57,65)(63,68)(64,67);(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,29)(13,34)(15,38)(16,39)(17,25)(18,40)(19,45)(21,46)(22,47)(23,48)(24,49)(30,54)(31,55)(32,56)(33,57)(35,59)(36,52)(37,60)(41,62)(42,63)(43,53)(44,64)(50,65)(51,66)(58,68)(61,67);(5,9,11,32,12,33,10,31)(6,15,16,42,18,44,17,43)(7,21,22,50,24,51,23,30)(8,26,28,56,29,57,27,55)(13,35,36,61,37,58,19,41)(14,38,39,63,40,64,25,53)(20,46,47,65,49,66,48,54)(34,59,52,67,60,68,45,62);(5,10,12,11)(6,16,18,17)(7,22,24,23)(8,27,29,28)(9,31,33,32)(13,19,37,36)(14,39,40,25)(15,42,44,43)(20,47,49,48)(21,50,51,30)(26,55,57,56)(34,45,60,52)(35,41,58,61)(38,63,64,53)(46,65,66,54)(59,62,68,67)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,f^b=fi,h^b=hi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289205);