Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #494749 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1932 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)(3,13,18,41)(4,19,22,39)(5,23,14,11)(6,12)(7,10)(8,16)(9,21,33,28)(15,44)(17,42,27,58)(20,53)(24,31)(25,52)(26,55,43,37)(29,35)(30,59,60,64)(32,49)(34,54)(36,57)(38,56)(40,50)(45,51,48,62)(46,61)(47,63);(1,3,14,16,7,18,5,24)(2,8,23,41,10,31,11,13)(4,15,20,48,22,47,25,45)(6,27,43,56,29,17,26,49)(9,30,34,61,33,60,36,40)(12,38,55,42,35,32,37,58)(19,51,52,44,39,62,53,63)(21,50,54,64,28,46,57,59);(1,4)(2,9)(3,15)(5,25)(6,28)(7,22)(8,30)(10,33)(11,36)(12,39)(13,40)(14,20)(16,48)(17,50)(18,47)(19,35)(21,29)(23,34)(24,45)(26,54)(27,46)(31,60)(32,51)(37,52)(38,62)(41,61)(42,63)(43,57)(44,58)(49,64)(53,55)(56,59);(2,10)(8,31)(9,33)(11,23)(12,35)(13,41)(19,39)(30,60)(32,38)(34,36)(37,55)(40,61)(42,58)(44,63)(51,62)(52,53)Polycyclic presentation:
2^9|a^2=g,b^2=eg,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303726, #72303735, #72303743, … |
| maximal normal subgroup of | #72303726, #72303735, #72303743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401945);