Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6458534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2504 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13,7,12,6,25)(3,14,17,31,18,9,26,38)(4,19,27,34,21,46,22,10)(5,11,28,36,23,37,24,49)(15,39,42,32,41,54,50,29)(16,35,44,55,43,30,51,58)(20,45,52,57,47,63,48,33)(40,59,61,56,60,62,64,53);(1,3,6,17,7,18,8,26)(2,9,25,38,12,14,13,31)(4,15,27,50,21,41,22,42)(5,16,28,51,23,43,24,44)(10,32,19,39,34,29,46,54)(11,35,36,58,37,30,49,55)(20,40,48,61,47,60,52,64)(33,53,63,59,57,56,45,62);(1,4,6,22,7,21,8,27)(2,10,13,19,12,34,25,46)(3,15,17,42,18,41,26,50)(5,20,24,48,23,47,28,52)(9,29,31,54,14,32,38,39)(11,33,36,45,37,57,49,63)(16,40,44,61,43,60,51,64)(30,53,55,62,35,56,58,59);(1,5)(2,11)(3,16)(4,20)(6,24)(7,23)(8,28)(9,30)(10,33)(12,37)(13,36)(14,35)(15,40)(17,44)(18,43)(19,45)(21,47)(22,48)(25,49)(26,51)(27,52)(29,53)(31,55)(32,56)(34,57)(38,58)(39,59)(41,60)(42,61)(46,63)(50,64)(54,62);(4,21)(10,34)(15,41)(19,46)(20,47)(22,27)(29,32)(33,57)(39,54)(40,60)(42,50)(45,63)(48,52)(53,56)(59,62)(61,64)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=g,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365730);