Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438013 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4478 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)(3,8)(4,17,20,9)(5,10)(6,12)(7,11)(13,29,44,55)(14,30)(15,31)(16,32)(18,33,46,48)(19,49,51,34)(21,41,26,35)(22,36)(23,38)(24,37)(25,40)(27,42)(28,39)(43,54,61,64)(45,56)(47,57)(50,58,62,63)(52,60,53,59);(1,3)(2,8)(4,18)(5,22)(6,25)(7,15)(9,33)(10,36)(11,31)(12,40)(13,21)(14,23)(16,28)(17,48)(19,43)(20,46)(24,47)(26,44)(27,45)(29,35)(30,38)(32,39)(34,54)(37,57)(41,55)(42,56)(49,64)(50,52)(51,61)(53,62)(58,59)(60,63);(1,4)(2,9)(3,13)(5,19)(6,26)(7,20)(8,29)(10,34)(11,17)(12,41)(14,43)(15,44)(16,46)(18,25)(21,28)(22,50)(23,53)(24,51)(27,52)(30,54)(31,55)(32,48)(33,40)(35,39)(36,58)(37,49)(38,60)(42,59)(45,61)(47,62)(56,64)(57,63);(1,5,7,24)(2,10,11,37)(3,14,15,45)(4,19,20,51)(6,27,28,23)(8,30,31,56)(9,34,17,49)(12,42,39,38)(13,43,44,61)(16,47,25,22)(18,50,46,62)(21,53,26,52)(29,54,55,64)(32,57,40,36)(33,58,48,63)(35,60,41,59);(2,11)(8,31)(9,17)(10,37)(12,39)(29,55)(30,56)(32,40)(33,48)(34,49)(35,41)(36,57)(38,42)(54,64)(58,63)(59,60)Polycyclic presentation:
2^9|a^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72303726, #72303739, #72303743, … |
| maximal normal subgroup of | #72303726, #72303739, #72303743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345210);