Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,32,45,5,29)(3,16,20,10,35,38,18,47)(4,23,26,42,24,44,25,34)(7,14,30,9,8,15,33,12)(11,41,43,27,57,59,60,28)(17,49,53,56,50,62,51,63)(19,40,21,37,31,39,22,48)(36,54,61,58,64,55,46,52);(2,9,14,13)(4,24)(5,30)(6,33)(10,16,39,37)(11,42,44,60)(12,15,29,45)(17,50)(18,31)(19,20)(23,43,57,34)(25,41)(26,59)(27,28)(36,61,63,49)(38,40,48,47)(46,56,62,64)(51,52)(53,58)(54,55);(1,3,8,22)(2,10,15,40)(4,17,28,55)(5,31,30,18)(6,19,33,20)(7,21,32,35)(9,16,29,48)(11,36,23,56)(12,38,13,37)(14,39,45,47)(24,50,27,54)(25,58,59,51)(26,52,41,53)(34,46,60,49)(42,61,43,62)(44,63,57,64);(1,4,7,27)(2,11,14,44)(3,17,21,54)(5,25,33,41)(6,26,30,59)(8,28,32,24)(9,34,13,43)(10,36,39,63)(12,42,29,60)(15,23,45,57)(16,46,37,62)(18,51,19,52)(20,53,31,58)(22,55,35,50)(38,61,48,49)(40,56,47,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105971);