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)(7,18)(8,24)(9,27)(10,31)(11,17)(12,14)(13,22)(15,30)(16,33)(19,39)(20,42)(21,38)(23,36)(25,40)(26,46)(28,58)(29,60)(32,61)(34,41)(35,44)(37,49)(43,56)(45,57)(47,52)(48,50)(51,65)(53,64)(54,66)(55,63)(59,68)(62,67);(1,2)(3,4)(5,7,11,23)(6,13,17,39)(8,19,12,22)(9,28,30,49)(10,21,34,54)(14,36,24,18)(15,42,27,47)(16,38,46,66)(20,35,52,25)(26,50,33,55)(29,59,32,62)(31,48,41,63)(37,40,58,44)(43,67,45,68)(51,57,53,56)(60,65,61,64);(2,4)(5,8,11,12)(6,14,17,24)(7,19,23,22)(9,25,30,35)(10,26,34,33)(13,36,39,18)(15,40,27,44)(16,41,46,31)(20,49,52,28)(21,50,54,55)(29,56,32,57)(37,42,58,47)(38,63,66,48)(43,60,45,61)(51,62,53,59)(64,67,65,68);(2,4)(5,9)(6,15)(7,20)(8,25)(10,32)(11,30)(12,35)(13,37)(14,40)(16,45)(17,27)(18,47)(19,49)(21,53)(22,28)(23,52)(24,44)(26,57)(29,34)(31,60)(33,56)(36,42)(38,65)(39,58)(41,61)(43,46)(48,67)(50,59)(51,54)(55,62)(63,68)(64,66);(5,10)(6,16)(7,21)(8,26)(9,29)(11,34)(12,33)(13,38)(14,41)(15,43)(17,46)(18,48)(19,50)(20,51)(22,55)(23,54)(24,31)(25,56)(27,45)(28,59)(30,32)(35,57)(36,63)(37,64)(39,66)(40,60)(42,67)(44,61)(47,68)(49,62)(52,53)(58,65)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,e^d=ei,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, 7421385);