Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3249 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)(3,13,15,33)(4,18,21,35)(5,22,25,10)(6,11)(7,12)(8,16)(9,20,32,26)(14,39)(17,38,43,62)(19,48)(23,29)(24,53,56,34)(27,36)(28,57,58,64)(30,44)(31,50)(37,45)(40,63)(41,46,49,60)(42,61)(47,51)(52,55)(54,59);(1,3)(2,8)(4,14)(5,23)(6,17)(7,15)(9,28)(10,33)(11,30)(12,29)(13,22)(16,25)(18,46)(19,49)(20,42)(21,40)(24,54)(26,45)(27,43)(31,61)(32,58)(34,62)(35,60)(36,59)(37,52)(38,53)(39,48)(41,51)(44,56)(47,63)(50,64)(55,57);(1,4)(2,9)(3,14)(5,19)(6,26)(7,21)(8,28)(10,31)(11,35)(12,32)(13,37)(15,40)(16,41)(17,45)(18,36)(20,27)(22,52)(23,49)(24,55)(25,51)(29,58)(30,60)(33,61)(34,48)(38,63)(39,62)(42,43)(44,64)(46,59)(47,53)(50,56)(54,57);(3,15)(8,29)(13,33)(14,40)(16,23)(17,43)(28,58)(30,59)(37,61)(38,62)(39,63)(41,49)(42,45)(44,54)(46,60)(57,64)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303730, #72303736, #72303748, … |
| maximal normal subgroup of | #72303730, #72303736, #72303748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400616);