Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6451734 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2404 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,19,41)(4,10,23,36)(5,11,27,40)(6,13,7,12)(9,28,32,18)(16,45,20,43)(17,44,24,53)(21,34,52,59)(22,42,29,35)(25,39,26,38)(30,47,33,48)(31,54,37,49)(46,62,50,61)(51,60,55,58)(56,63,57,64);(1,3)(2,9)(4,20)(5,24)(6,18)(7,28)(8,19)(10,33)(11,37)(12,15)(13,41)(14,32)(16,23)(17,27)(21,50)(22,48)(25,54)(26,49)(29,47)(30,36)(31,40)(34,57)(35,45)(38,53)(39,44)(42,43)(46,52)(51,64)(55,63)(56,59)(58,62)(60,61);(1,4)(2,10)(3,16,19,20)(5,21)(6,22)(7,29)(8,23)(9,30,32,33)(11,34)(12,35)(13,42)(14,36)(15,43,41,45)(17,46,24,50)(18,47,28,48)(25,51)(26,55)(27,52)(31,56,37,57)(38,58)(39,60)(40,59)(44,61,53,62)(49,63,54,64);(1,5)(2,11)(3,17,19,24)(4,21)(6,25)(7,26)(8,27)(9,31,32,37)(10,34)(12,38)(13,39)(14,40)(15,44,41,53)(16,46,20,50)(18,49,28,54)(22,51)(23,52)(29,55)(30,56,33,57)(35,58)(36,59)(42,60)(43,61,45,62)(47,63,48,64);(1,6,8,7)(2,12,14,13)(3,18,19,28)(4,22,23,29)(5,25,27,26)(9,15,32,41)(10,35,36,42)(11,38,40,39)(16,47,20,48)(17,49,24,54)(21,51,52,55)(30,43,33,45)(31,44,37,53)(34,58,59,60)(46,63,50,64)(56,61,57,62)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358933);