Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14020 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,20,32,19,9,18,33)(4,21,24,35,23,38,29,10)(5,25,27,36,28,11,26,37)(16,41,44,30,43,55,48,39)(17,31,47,57,46,40,45,56)(22,49,51,34,50,58,52,53)(42,60,62,59,61,64,63,54);(1,3)(2,9)(4,16,23,43)(5,17)(6,20)(7,18)(8,19)(10,30,35,39)(11,31)(12,15)(13,33)(14,32)(21,41,38,55)(22,42,50,61)(24,48,29,44)(25,40)(26,47)(27,45)(28,46)(34,54,53,59)(36,57)(37,56)(49,60,58,64)(51,63,52,62);(1,4)(2,10,12,35)(3,16,19,43)(5,22)(6,24)(7,29)(8,23)(9,30)(11,34,25,53)(13,21,14,38)(15,39)(17,42,46,61)(18,44,20,48)(26,51)(27,52)(28,50)(31,54)(32,55)(33,41)(36,58,37,49)(40,59)(45,62,47,63)(56,60)(57,64);(1,5,8,28)(2,11,12,25)(3,17,19,46)(4,22,23,50)(6,27,7,26)(9,31,15,40)(10,34,35,53)(13,37,14,36)(16,42,43,61)(18,47,20,45)(21,49,38,58)(24,52,29,51)(30,54,39,59)(32,57,33,56)(41,60,55,64)(44,63,48,62)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=eg,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275750);