Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,10,15,11,16,24,37)(7,17,20,34,21,43,31,12)(8,22,27,51,26,13,35,39)(9,14,29,41,30,42,54,62)(18,46,48,32,23,44,53,36)(19,45,49,40,28,57,58,33)(25,50,56,67,55,38,60,63)(47,65,66,59,52,64,68,61);(5,7,11,21)(6,12,16,34)(8,23,26,18)(9,28,30,19)(10,31,24,20)(13,36,22,32)(14,40,42,33)(15,43,37,17)(25,52,55,47)(27,48,35,53)(29,49,54,58)(38,61,50,59)(39,44,51,46)(41,45,62,57)(56,66,60,68)(63,64,67,65);(1,2,3,4)(5,8,24,35,11,26,10,27)(6,13,37,51,16,22,15,39)(7,18,31,53,21,23,20,48)(9,25,54,60,30,55,29,56)(12,32,43,46,34,36,17,44)(14,38,62,67,42,50,41,63)(19,47,58,68,28,52,49,66)(33,59,57,65,40,61,45,64);(5,9,10,29,11,30,24,54)(6,14,15,41,16,42,37,62)(7,19,20,49,21,28,31,58)(8,25,27,56,26,55,35,60)(12,33,17,45,34,40,43,57)(13,38,39,63,22,50,51,67)(18,47,48,66,23,52,53,68)(32,59,44,64,36,61,46,65);(8,26)(13,22)(18,23)(25,55)(27,35)(32,36)(38,50)(39,51)(44,46)(47,52)(48,53)(56,60)(59,61)(63,67)(64,65)(66,68)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=f,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351523);