Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1879 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1879 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,27,45,9,15,6,29,5,12,28,42)(3,16,20,37,18,43,32,38,21,49,19,46,34,59,30,10)(4,22,25,47,23,58,33,11,26,40,24,39,35,44,31,41)(17,36,52,61,53,63,56,48,54,64,51,62,50,60,55,57);(1,3)(2,10,15,38)(4,17)(5,18)(6,30)(7,32)(8,34)(9,21)(11,36,41,64)(12,37,14,46)(13,43,29,59)(16,42,49,45)(19,28)(20,27)(22,57,40,48)(23,50)(24,55)(25,56)(26,54)(31,51)(33,52)(35,53)(39,60,47,63)(44,61,58,62);(1,4)(2,11)(3,17)(5,23)(6,31)(7,33)(8,35)(9,26)(10,36)(12,39)(13,44)(14,47)(15,41)(16,48)(18,50)(19,55)(20,56)(21,54)(22,42)(24,28)(25,27)(29,58)(30,51)(32,52)(34,53)(37,60)(38,64)(40,45)(43,61)(46,63)(49,57)(59,62);(1,5,9,8)(2,12,15,14)(3,18,21,34)(4,23,26,35)(6,27,7,28)(10,37,38,46)(11,39,41,47)(13,42,29,45)(16,43,49,59)(17,50,54,53)(19,30,20,32)(22,58,40,44)(24,31,25,33)(36,60,64,63)(48,61,57,62)(51,56,52,55)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279896, #72280389, #72281128, … |
| maximal normal subgroup of | #72279896, #72280389, #72281128, … |
| maximal quotient of | C2×#494547, #72310147, #72342567 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401743);