Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6371542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#386 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,19,31,18,39,17,9)(4,20,23,36,24,10,22,33)(5,11,27,35,25,34,26,52)(15,37,43,29,42,54,41,47)(16,38,46,55,44,59,45,30)(21,48,51,57,49,32,50,56)(40,58,62,53,60,64,61,63);(1,3)(2,9,13,31)(4,15)(5,16)(6,17)(7,19)(8,18)(10,29,20,47)(11,30,34,55)(12,14,28,39)(21,40)(22,41)(23,43)(24,42)(25,44)(26,45)(27,46)(32,53,48,63)(33,54,36,37)(35,38,52,59)(49,60)(50,61)(51,62)(56,64,57,58);(1,4,8,24)(2,10,13,20)(3,15,18,42)(5,21,25,49)(6,23,7,22)(9,29,31,47)(11,32,34,48)(12,36,28,33)(14,37,39,54)(16,40,44,60)(17,43,19,41)(26,51,27,50)(30,53,55,63)(35,57,52,56)(38,58,59,64)(45,62,46,61);(1,5)(2,11)(3,16)(4,21)(6,26)(7,27)(8,25)(9,30)(10,32)(12,35)(13,34)(14,38)(15,40)(17,45)(18,44)(19,46)(20,48)(22,50)(23,51)(24,49)(28,52)(29,53)(31,55)(33,56)(36,57)(37,58)(39,59)(41,61)(42,60)(43,62)(47,63)(54,64);(5,25)(11,34)(16,44)(21,49)(26,27)(30,55)(32,48)(35,52)(38,59)(40,60)(45,46)(50,51)(53,63)(56,57)(58,64)(61,62)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,g^b=gi,f^c=fi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278744);