Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4410 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)(7,19,22,38)(8,25,30,41)(9,15,34,46)(10,17,11,16)(13,24,37,23)(14,28,42,29)(20,35,48,59)(21,43,50,61)(26,53,49,39)(27,52,57,63)(31,36,51,60)(32,45,33,44)(40,56,64,55)(47,58,65,68)(54,67,66,62);(2,4)(5,7)(6,13)(8,26)(9,31)(10,24)(11,23)(12,22)(14,39)(15,43)(16,38)(17,19)(18,37)(20,29)(21,33)(25,35)(27,47)(28,48)(30,49)(32,50)(34,51)(36,45)(40,58)(41,59)(42,53)(44,60)(46,61)(52,67)(54,56)(55,66)(57,65)(62,63)(64,68);(5,8,11,28,12,30,10,29)(6,14,17,41,18,42,16,25)(7,20,23,26,22,48,24,49)(9,27,32,56,34,57,33,55)(13,35,19,39,37,59,38,53)(15,40,44,63,46,64,45,52)(21,47,31,54,50,65,51,66)(36,58,43,62,60,68,61,67);(5,9)(6,15)(7,21)(8,27)(10,33)(11,32)(12,34)(13,36)(14,40)(16,45)(17,44)(18,46)(19,43)(20,47)(22,50)(23,31)(24,51)(25,52)(26,54)(28,56)(29,55)(30,57)(35,58)(37,60)(38,61)(39,62)(41,63)(42,64)(48,65)(49,66)(53,67)(59,68);(7,22)(13,37)(19,38)(20,48)(21,50)(23,24)(26,49)(31,51)(35,59)(36,60)(39,53)(43,61)(47,65)(54,66)(58,68)(62,67)Polycyclic presentation:
2^9|a^2=i,c^2=gi,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,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349908);