Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2352 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13885 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
705 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)(3,14)(4,19,23,10)(5,24)(6,28)(7,12)(8,13)(9,17)(11,25)(15,41,44,35)(16,45)(18,34)(20,31,43,58)(21,50,52,53)(22,37,30,38)(26,39)(27,54)(29,33)(32,47)(36,51,48,56)(40,46)(42,57,63,64)(49,60,62,61)(55,59);(1,3)(2,9,13,33)(4,20)(5,16)(6,17)(7,29)(8,18)(10,35,19,41)(11,32,39,45)(12,14,28,34)(15,22)(21,49)(23,43)(24,40,54,59)(25,46)(26,55)(27,47)(30,44)(31,37,58,38)(36,60,48,61)(42,51)(50,57,53,64)(52,62)(56,63);(1,4,6,22,8,23,7,30)(2,10,28,38,13,19,12,37)(3,15,17,43,18,44,29,20)(5,21,25,51,27,52,26,56)(9,31,34,41,33,58,14,35)(11,36,54,50,39,48,24,53)(16,42,46,62,47,63,55,49)(32,57,59,61,45,64,40,60);(1,5,6,25,8,27,7,26)(2,11,12,24,13,39,28,54)(3,16,17,46,18,47,29,55)(4,21,22,51,23,52,30,56)(9,32,14,40,33,45,34,59)(10,36,37,53,19,48,38,50)(15,42,43,62,44,63,20,49)(31,57,35,60,58,64,41,61)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268168);