Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| 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 | |
4464 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)(5,6,10,17,12,16,11,18)(7,19,22,37,24,38,23,13)(8,25,28,41,30,42,29,14)(9,15,32,45,33,44,34,46)(20,35,48,60,49,52,26,39)(21,43,31,58,50,61,51,36)(27,53,55,63,56,64,57,40)(47,59,65,68,66,67,54,62);(2,4)(5,7,12,24)(6,13,16,37)(8,26,30,48)(9,31,33,51)(10,23,11,22)(14,39,41,60)(15,43,44,61)(17,38,18,19)(20,28,49,29)(21,34,50,32)(25,52,42,35)(27,47,56,66)(36,46,58,45)(40,59,63,67)(53,62,64,68)(54,57,65,55);(5,8,12,30)(6,14,16,41)(7,20,24,49)(9,27,33,56)(10,29,11,28)(13,35,37,52)(15,40,44,63)(17,42,18,25)(19,39,38,60)(21,47,50,66)(22,26,23,48)(31,54,51,65)(32,57,34,55)(36,59,58,67)(43,62,61,68)(45,64,46,53);(5,9)(6,15)(7,21)(8,27)(10,32)(11,34)(12,33)(13,36)(14,40)(16,44)(17,45)(18,46)(19,43)(20,47)(22,31)(23,51)(24,50)(25,53)(26,54)(28,55)(29,57)(30,56)(35,59)(37,58)(38,61)(39,62)(41,63)(42,64)(48,65)(49,66)(52,67)(60,68);(6,16)(13,37)(14,41)(15,44)(17,18)(19,38)(25,42)(35,52)(36,58)(39,60)(40,63)(43,61)(45,46)(53,64)(59,67)(62,68)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,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, 6345102);