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 | |
1239 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)(4,19,23,34)(5,24)(6,12)(7,13)(8,17)(9,22,35,47)(10,25)(11,27)(15,28,20,32)(16,42)(18,30)(21,33,51,58)(26,36)(29,44)(31,45)(37,53)(38,41,55,49)(39,43)(40,59,48,63)(46,62,52,50)(54,64,57,60)(56,61);(1,3,6,17,7,18,27,45)(2,8,11,14,13,31,12,30)(4,20,47,41,23,15,22,49)(5,16,25,43,26,44,53,61)(9,32,19,38,35,28,34,55)(10,29,24,39,37,42,36,56)(21,48,62,60,51,40,50,64)(33,57,46,59,58,54,52,63);(1,4,7,23)(2,9)(3,15,18,20)(5,21,26,51)(6,22,27,47)(8,28)(10,33)(11,34)(12,19)(13,35)(14,38)(16,40,44,48)(17,41,45,49)(24,52)(25,50,53,62)(29,54)(30,55)(31,32)(36,46)(37,58)(39,59)(42,57)(43,60,61,64)(56,63);(1,5,6,25,7,26,27,53)(2,10,11,24,13,37,12,36)(3,16,17,43,18,44,45,61)(4,21,22,50,23,51,47,62)(8,29,14,39,31,42,30,56)(9,33,34,52,35,58,19,46)(15,40,41,60,20,48,49,64)(28,54,38,59,32,57,55,63)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,e^2=i,f^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, 272965);