Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #390446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14572 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,12,6,13)(3,15,20,32,19,9,18,33)(4,21,24,10,25,37,23,36)(5,26,27,11,28,40,29,39)(16,41,44,34,45,50,43,30)(17,46,47,38,48,56,49,31)(22,52,53,51,54,35,55,57)(42,58,61,60,62,59,63,64);(1,3,7,19)(2,9,12,15)(4,16,25,45)(5,17,28,48)(6,20,8,18)(10,34,36,30)(11,38,39,31)(13,33,14,32)(21,50,37,41)(22,42,54,62)(23,44,24,43)(26,56,40,46)(27,49,29,47)(35,58,52,59)(51,60,57,64)(53,63,55,61);(1,4,6,24,7,25,8,23)(2,10,13,37,12,36,14,21)(3,16,18,44,19,45,20,43)(5,22,29,53,28,54,27,55)(9,30,32,41,15,34,33,50)(11,35,26,57,39,52,40,51)(17,42,49,61,48,62,47,63)(31,58,56,60,38,59,46,64);(1,5)(2,11)(3,17)(4,22)(6,29)(7,28)(8,27)(9,31)(10,35)(12,39)(13,26)(14,40)(15,38)(16,42)(18,49)(19,48)(20,47)(21,51)(23,55)(24,53)(25,54)(30,58)(32,56)(33,46)(34,59)(36,52)(37,57)(41,60)(43,63)(44,61)(45,62)(50,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=egh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297645);