Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3029 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)(3,4)(5,6)(7,19)(8,25)(9,29)(10,18)(11,17)(12,23)(13,28)(14,31)(15,30)(16,34)(20,47)(21,46)(22,36)(24,38)(26,39)(27,41)(32,44)(33,37)(35,60)(40,59)(42,61)(43,53)(45,51)(48,64)(49,68)(50,54)(52,63)(55,58)(56,57)(62,66)(65,67);(1,2)(3,4)(5,7,11,24)(6,12,17,37)(8,20,28,51)(9,22,31,52)(10,33,34,23)(13,35,25,54)(14,43,29,46)(15,36,44,63)(16,38,18,19)(21,30,53,32)(26,49,58,66)(27,60,61,50)(39,64,55,65)(40,62,56,68)(41,47,42,45)(48,57,67,59);(2,4)(5,8,11,28)(6,13,17,25)(7,20,24,51)(9,26,31,58)(10,27,34,61)(12,35,37,54)(14,39,29,55)(15,40,44,56)(16,41,18,42)(19,45,38,47)(21,48,53,67)(22,49,52,66)(23,50,33,60)(30,57,32,59)(36,62,63,68)(43,64,46,65);(2,4)(5,9)(6,14)(7,21)(8,26)(10,30)(11,31)(12,36)(13,39)(15,16)(17,29)(18,44)(19,43)(20,48)(22,33)(23,52)(24,53)(25,55)(27,57)(28,58)(32,34)(35,62)(37,63)(38,46)(40,41)(42,56)(45,64)(47,65)(49,60)(50,66)(51,67)(54,68)(59,61);(5,9)(6,15)(7,22)(8,26)(10,32)(11,31)(12,36)(13,40)(14,16)(17,44)(18,29)(19,46)(20,49)(21,33)(23,53)(24,52)(25,56)(27,59)(28,58)(30,34)(35,62)(37,63)(38,43)(39,41)(42,55)(45,65)(47,64)(48,60)(50,67)(51,66)(54,68)(57,61)Polycyclic presentation:
2^9|b^2=i,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421373);