Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442596 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2487 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,12,7,13,6,26)(3,9,27,33,16,32,17,42)(4,18,28,35,21,44,22,10)(5,11,29,37,24,36,25,52)(14,38,47,55,19,45,40,30)(15,31,51,56,23,50,41,60)(20,43,53,57,48,61,49,34)(39,58,63,64,46,62,59,54);(1,3)(2,9)(4,19)(5,23)(6,17)(7,16)(8,27)(10,30)(11,31)(12,33)(13,32)(14,21)(15,24)(18,38)(20,39)(22,47)(25,51)(26,42)(28,40)(29,41)(34,54)(35,55)(36,50)(37,56)(43,58)(44,45)(46,48)(49,59)(52,60)(53,63)(57,64)(61,62);(1,4,6,22,7,21,8,28)(2,10,12,18,13,35,26,44)(3,14,17,40,16,19,27,47)(5,20,25,49,24,48,29,53)(9,30,33,38,32,55,42,45)(11,34,37,43,36,57,52,61)(15,39,41,59,23,46,51,63)(31,54,56,58,50,64,60,62);(1,5)(2,11)(3,15)(4,20)(6,25)(7,24)(8,29)(9,31)(10,34)(12,37)(13,36)(14,39)(16,23)(17,41)(18,43)(19,46)(21,48)(22,49)(26,52)(27,51)(28,53)(30,54)(32,50)(33,56)(35,57)(38,58)(40,59)(42,60)(44,61)(45,62)(47,63)(55,64);(3,16)(4,21)(9,32)(10,35)(15,23)(17,27)(18,44)(20,48)(22,28)(31,50)(33,42)(34,57)(41,51)(43,61)(49,53)(56,60)Polycyclic presentation:
2^9|a^2=gh,c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365417);