Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #366639 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13995 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2171 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)(3,13,18,8)(4,19)(5,23,25,10)(6,26)(7,11)(9,22)(12,14)(15,41,43,55)(16,30)(17,31,40,39)(20,49,50,59)(21,36)(24,35,28,37)(27,34)(29,44,32,47)(33,51,53,52)(38,46)(42,60)(45,57)(48,56)(54,62)(58,63)(61,64);(1,3,14,40,7,18,6,17)(2,8,12,31,11,13,26,39)(4,15,27,47,21,43,22,44)(5,16,24,45,25,46,28,48)(9,32,19,41,34,29,36,55)(10,30,37,57,23,38,35,56)(20,42,51,62,50,61,52,63)(33,58,49,60,53,54,59,64);(1,4)(2,9)(3,15)(5,20)(6,27)(7,21)(8,29)(10,33)(11,34)(12,36)(13,32)(14,22)(16,42)(17,47)(18,43)(19,26)(23,53)(24,52)(25,50)(28,51)(30,54)(31,41)(35,49)(37,59)(38,58)(39,55)(40,44)(45,63)(46,61)(48,62)(56,64)(57,60);(1,5)(2,10)(3,16)(4,20)(6,28)(7,25)(8,30)(9,33)(11,23)(12,37)(13,38)(14,24)(15,42)(17,48)(18,46)(19,49)(21,50)(22,52)(26,35)(27,51)(29,54)(31,57)(32,58)(34,53)(36,59)(39,56)(40,45)(41,60)(43,61)(44,63)(47,62)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72341342, #72341344, #72341363, … |
| maximal normal subgroup of | #72341342, #72341344, #72341363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273835);