Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6928 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)(5,6,12,18)(7,19,24,36)(8,25,30,40)(9,15,31,42)(10,17,11,16)(13,23,37,22)(14,29,41,28)(20,46,49,34)(21,45,50,61)(26,38,48,59)(27,53,55,64)(32,44,33,43)(35,52,60,51)(39,57,63,56)(47,65,66,58)(54,62,67,68);(2,4)(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,38)(15,35)(16,36)(17,19)(18,37)(20,29)(25,46)(27,54)(28,49)(30,48)(31,50)(32,52)(33,51)(34,40)(39,62)(41,59)(42,60)(43,61)(44,45)(47,57)(53,65)(55,67)(56,66)(58,64)(63,68);(5,8,12,30)(6,14,18,41)(7,20,24,49)(9,27,31,55)(10,29,11,28)(13,34,37,46)(15,39,42,63)(16,40,17,25)(19,38,36,59)(21,47,50,66)(22,26,23,48)(32,57,33,56)(35,58,60,65)(43,64,44,53)(45,62,61,68)(51,54,52,67);(5,9)(6,15)(7,21)(8,27)(10,32)(11,33)(12,31)(13,35)(14,39)(16,43)(17,44)(18,42)(19,45)(20,47)(22,51)(23,52)(24,50)(25,53)(26,54)(28,56)(29,57)(30,55)(34,58)(36,61)(37,60)(38,62)(40,64)(41,63)(46,65)(48,67)(49,66)(59,68);(9,31)(15,42)(21,50)(27,55)(32,33)(35,60)(39,63)(43,44)(45,61)(47,66)(51,52)(53,64)(54,67)(56,57)(58,65)(62,68)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338983);