Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12750 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1786 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1786 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3129 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)(3,14,18,42)(4,20,24,33)(5,10)(6,27)(7,13)(8,17,32,49)(9,23,35,21)(11,25)(12,26)(15,43)(16,41,19,37)(22,54,57,34)(28,39,50,31)(29,36)(30,58)(38,48)(40,44)(45,59)(46,63)(47,62)(51,64)(52,56,61,53)(55,60);(1,3)(2,8)(4,15)(5,16)(6,28)(7,18)(9,30)(10,31)(11,37)(12,39)(13,32)(14,36)(17,48)(19,26)(20,51)(21,44)(22,55)(23,46)(24,47)(25,49)(27,42)(29,50)(33,59)(34,43)(35,60)(38,41)(40,61)(45,56)(52,63)(53,64)(54,62)(57,58);(1,4)(2,9)(3,15)(5,21)(6,22)(7,24)(8,30)(10,33)(11,34)(12,20)(13,35)(14,40)(16,44)(17,45)(18,47)(19,46)(23,26)(25,53)(27,52)(28,55)(29,57)(31,59)(32,60)(36,61)(37,43)(38,54)(39,51)(41,62)(42,63)(48,56)(49,64)(50,58);(1,5,7,26)(2,10,13,12)(3,16,18,19)(4,21,24,23)(6,25,29,48)(8,31,32,39)(9,33,35,20)(11,36,38,27)(14,41,42,37)(15,44,47,46)(17,28,49,50)(22,53,57,56)(30,59,60,51)(34,61,54,52)(40,62,63,43)(45,55,64,58)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280034, #72280389, #72280993, … |
| maximal normal subgroup of | #72280034, #72280389, #72280993, … |
| maximal quotient of | C2×#493556, #72310144, #72321298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400752);