Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13998 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,29)(3,14,18,33,19,32,17,9)(4,20,23,10,24,36,22,37)(5,25,28,38,27,54,26,11)(15,34,43,57,44,30,42,39)(16,45,47,56,48,31,46,40)(21,35,51,53,52,59,50,49)(41,61,63,55,64,60,62,58);(1,3,8,18,7,19,6,17)(2,9,29,32,13,33,12,14)(4,15,22,42,24,44,23,43)(5,16,28,47,27,48,26,46)(10,34,36,57,37,30,20,39)(11,31,54,56,38,45,25,40)(21,41,50,62,52,64,51,63)(35,58,53,61,59,55,49,60);(1,4)(2,10)(3,15)(5,21)(6,22)(7,24)(8,23)(9,30)(11,35)(12,36)(13,37)(14,39)(16,41)(17,42)(18,43)(19,44)(20,29)(25,53)(26,50)(27,52)(28,51)(31,55)(32,57)(33,34)(38,59)(40,60)(45,58)(46,62)(47,63)(48,64)(49,54)(56,61);(1,5,6,26,7,27,8,28)(2,11,12,25,13,38,29,54)(3,16,17,46,19,48,18,47)(4,21,22,50,24,52,23,51)(9,31,14,40,33,45,32,56)(10,35,36,53,37,59,20,49)(15,41,42,62,44,64,43,63)(30,55,39,60,34,58,57,61)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274296);