Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500730 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1032 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)(7,21)(8,28)(9,19)(10,36)(11,16)(12,40)(13,20)(14,24)(15,32)(17,37)(18,35)(22,58)(23,45)(25,44)(26,47)(27,46)(29,66)(30,52)(31,54)(33,56)(34,50)(38,55)(39,49)(41,53)(42,51)(43,63)(48,64)(57,61)(59,65)(60,67)(62,68);(5,7,10,24,13,27,37,26)(6,14,17,21,20,47,36,46)(8,29,31,59,34,61,33,63)(9,23,11,38,35,25,12,41)(15,48,49,68,52,67,51,58)(16,44,19,55,40,45,18,53)(22,39,60,32,62,42,64,30)(28,65,56,66,50,43,54,57);(2,4)(5,8)(6,15)(7,22)(9,30)(10,31)(11,39)(12,42)(13,34)(14,43)(16,28)(17,49)(18,54)(19,56)(20,52)(21,57)(23,59)(24,60)(25,63)(26,64)(27,62)(29,41)(32,35)(33,37)(36,51)(38,61)(40,50)(44,58)(45,68)(46,66)(47,65)(48,55)(53,67);(5,9,10,11,13,35,37,12)(6,16,17,19,20,40,36,18)(7,23,24,38,27,25,26,41)(8,30,31,39,34,32,33,42)(14,44,21,55,47,45,46,53)(15,28,49,56,52,50,51,54)(22,59,60,61,62,63,64,29)(43,58,57,48,65,68,66,67)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407926);