Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,28)(3,9,17,32,18,31,16,46)(4,19,24,38,22,10,23,35)(5,11,27,36,25,37,26,53)(14,39,42,47,41,29,20,33)(15,30,45,55,43,56,44,63)(21,34,52,58,50,48,51,59)(40,54,62,57,61,60,49,64);(1,3,7,17,8,18,6,16)(2,9,13,32,12,31,28,46)(4,20,24,14,22,42,23,41)(5,15,27,45,25,43,26,44)(10,33,35,39,19,47,38,29)(11,30,36,55,37,56,53,63)(21,49,52,40,50,62,51,61)(34,57,58,60,48,64,59,54);(1,4)(2,10)(3,14)(5,21)(6,24)(7,23)(8,22)(9,29)(11,34)(12,19)(13,38)(15,40)(16,42)(17,20)(18,41)(25,50)(26,52)(27,51)(28,35)(30,54)(31,39)(32,47)(33,46)(36,59)(37,48)(43,61)(44,62)(45,49)(53,58)(55,64)(56,60)(57,63);(1,5)(2,11)(3,15)(4,21)(6,26)(7,27)(8,25)(9,30)(10,34)(12,37)(13,36)(14,40)(16,44)(17,45)(18,43)(19,48)(20,49)(22,50)(23,51)(24,52)(28,53)(29,54)(31,56)(32,55)(33,57)(35,58)(38,59)(39,60)(41,61)(42,62)(46,63)(47,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271727);