Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6371542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#386 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,18,21,9)(5,10)(6,24)(7,12)(8,16)(11,26)(14,35,40,53)(15,36)(17,37)(19,46,47,30)(20,31,25,34)(22,32)(23,50)(27,39,45,43)(28,42)(29,44)(33,49)(38,57,59,64)(41,58)(48,55,51,56)(52,60,63,62)(54,61);(1,3,6,16,7,17,26,44)(2,8,11,13,12,29,24,37)(4,14,25,43,21,40,20,39)(5,15,23,42,22,41,49,61)(9,27,34,35,18,45,31,53)(10,28,33,36,32,54,50,58)(19,38,51,62,47,59,48,60)(30,52,56,57,46,63,55,64);(1,4)(2,9)(3,14)(5,19)(6,25)(7,21)(8,27)(10,30)(11,34)(12,18)(13,35)(15,38)(16,43)(17,40)(20,26)(22,47)(23,51)(24,31)(28,52)(29,45)(32,46)(33,56)(36,57)(37,53)(39,44)(41,59)(42,62)(48,49)(50,55)(54,63)(58,64)(60,61);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,28)(9,30)(11,33)(12,32)(13,36)(14,38)(16,42)(17,41)(18,46)(20,48)(21,47)(24,50)(25,51)(26,49)(27,52)(29,54)(31,55)(34,56)(35,57)(37,58)(39,60)(40,59)(43,62)(44,61)(45,63)(53,64);(5,22)(10,32)(15,41)(19,47)(23,49)(28,54)(30,46)(33,50)(36,58)(38,59)(42,61)(48,51)(52,63)(55,56)(57,64)(60,62)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,g^b=gi,f^c=fi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72307021, #72307024, #72307049, … |
| maximal normal subgroup of | #72307021, #72307024, #72307049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278738);