Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C4×B(4,𝔽2) |
| prime radical | #33550 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,24,35)(3,13,17,38,14,36,40,60)(4,9,21,32,19,30,47,57)(5,10,23,34,22,33,49,59)(8,25,28,39,26,51,52,16)(15,37,42,61,18,43,45,62)(20,31,48,58,46,56,63,64)(27,50,53,44,29,54,55,41);(1,3)(2,8)(4,18)(5,16)(6,17)(7,14)(9,29)(10,13)(11,28)(12,26)(15,19)(20,44)(21,45)(22,39)(23,25)(24,40)(27,30)(31,43)(32,55)(33,36)(34,38)(35,52)(37,56)(41,46)(42,47)(48,54)(49,51)(50,63)(53,57)(58,62)(59,60)(61,64);(3,14)(4,19)(5,22)(8,26)(9,30)(13,36)(17,40)(21,47)(23,49)(28,52)(31,56)(32,57)(38,60)(41,44)(50,54)(58,64);(1,4,6,21,7,19,24,47)(2,9,11,32,12,30,35,57)(3,15,17,42,14,18,40,45)(5,20,23,48,22,46,49,63)(8,27,28,53,26,29,52,55)(10,31,34,58,33,56,59,64)(13,37,38,61,36,43,60,62)(16,41,25,50,39,44,51,54)Polycyclic presentation:
2^9|a^2=g,d^2=g,g^2=i,b^a=be,c^a=cf,h^a=hi,c^b=ci,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312376, #72312381, #72312401, … |
| maximal normal subgroup of | #72312376, #72312381, #72312401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402952);