Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1853 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1853 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1017 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)(3,15,19,43)(4,20)(5,11)(6,26)(7,13)(8,14)(9,18,34,49)(10,23)(12,28)(16,44,48,61)(17,42,30,37)(21,36)(22,53)(24,40)(25,39)(27,33,57,32)(29,38)(31,58,59,54)(35,56)(41,45,63,47)(46,60,64,50)(51,55)(52,62);(1,3,7,30,8,19,5,17)(2,9,11,32,14,34,13,33)(4,16,23,47,24,48,21,45)(6,27,28,18,29,57,25,49)(10,31,20,50,36,59,40,60)(12,37,26,15,39,42,38,43)(22,54,55,46,56,58,52,64)(35,61,51,41,53,44,62,63);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,31)(11,20)(12,35)(13,40)(14,36)(15,41)(17,45)(18,46)(19,48)(25,52)(26,51)(27,54)(28,55)(29,56)(30,47)(32,50)(33,60)(34,59)(37,61)(38,62)(39,53)(42,44)(43,63)(49,64)(57,58);(1,5,8,7)(2,11,14,13)(3,17,19,30)(4,21,24,23)(6,25,29,28)(9,32,34,33)(10,20,36,40)(12,26,39,38)(15,42,43,37)(16,45,48,47)(18,27,49,57)(22,52,56,55)(31,50,59,60)(35,51,53,62)(41,44,63,61)(46,54,64,58)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gi,h^b=hi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285601, #72285775, #72285805, … |
| maximal normal subgroup of | #72285601, #72285775, #72285805, … |
| maximal quotient of | #72310107, #72321396, #72322830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401471);