Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2622 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2266 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)(3,14)(4,19)(5,11)(6,12)(7,28)(8,13)(9,17)(10,24)(15,41)(16,40)(18,32)(20,39)(21,35)(22,47)(23,36)(25,37)(26,53)(27,38)(29,33)(30,49)(31,44)(34,43)(42,60)(45,56)(46,52)(48,61)(50,59)(51,63)(54,57)(55,62)(58,64);(1,3)(2,9)(4,20)(5,16)(6,17)(7,29)(8,18)(10,34)(11,31)(12,32)(13,33)(14,28)(15,24)(19,30)(21,48)(22,49)(23,43)(25,44)(26,54)(27,45)(35,58)(36,39)(37,56)(38,57)(40,53)(41,47)(42,52)(46,55)(50,64)(51,62)(59,60)(61,63);(1,4,7,23,8,24,6,22)(2,10,12,36,13,19,28,47)(3,15,17,43,18,20,29,49)(5,21,26,51,27,52,25,50)(9,30,14,39,33,34,32,41)(11,35,37,59,38,46,53,63)(16,42,44,62,45,48,54,64)(31,55,40,60,57,58,56,61);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,27)(9,31)(10,35)(12,37)(13,38)(14,40)(15,42)(17,44)(18,45)(19,46)(20,48)(22,50)(23,51)(24,52)(28,53)(29,54)(30,55)(32,56)(33,57)(34,58)(36,59)(39,60)(41,61)(43,62)(47,63)(49,64);(1,6,8,7)(2,12,13,28)(3,17,18,29)(4,22,24,23)(5,25,27,26)(9,32,33,14)(10,36,19,47)(11,37,38,53)(15,43,20,49)(16,44,45,54)(21,50,52,51)(30,41,34,39)(31,56,57,40)(35,59,46,63)(42,62,48,64)(55,61,58,60)Polycyclic presentation:
2^9|c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340368);