Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #365053 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13979 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,18,33,19,41,17,9)(4,20,23,10,25,36,24,37)(5,26,28,38,29,56,27,11)(15,34,43,39,21,31,44,49)(16,45,47,61,48,32,46,40)(22,52,53,50,55,35,54,59)(42,62,63,58,51,60,64,57);(1,3)(2,9,13,33)(4,21,25,15)(5,16)(6,17)(7,18)(8,19)(10,34)(11,32,38,45)(12,14,30,41)(20,49)(22,51,55,42)(23,44,24,43)(26,40,56,61)(27,46)(28,47)(29,48)(31,37)(35,58)(36,39)(50,62)(52,57)(53,64,54,63)(59,60);(1,4)(2,10,13,37)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31,33,34)(11,35,38,52)(12,20,30,36)(14,39,41,49)(16,42)(17,43)(18,44)(19,21)(26,50,56,59)(27,53)(28,54)(29,55)(32,57,45,58)(40,60,61,62)(46,63)(47,64)(48,51);(1,5,6,27,8,29,7,28)(2,11,12,26,13,38,30,56)(3,16,17,46,19,48,18,47)(4,22,23,53,25,55,24,54)(9,32,14,40,33,45,41,61)(10,35,20,50,37,52,36,59)(15,42,43,63,21,51,44,64)(31,57,39,60,34,58,49,62)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272966);