Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374204 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4205 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)(3,15,20,9)(4,21,24,39)(5,25,28,11)(6,12)(7,14)(8,13)(10,23,36,29)(16,41)(17,31)(18,33,19,32)(22,51)(26,38,27,37)(30,43)(34,48)(35,52)(40,47)(42,61,63,60)(44,49)(45,57)(46,58)(50,53)(54,55)(56,62,59,64);(1,3,8,20)(2,9,13,15)(4,16,24,44)(5,17,28,47)(6,19,7,18)(10,34,36,30)(11,31,25,40)(12,33,14,32)(21,49,39,41)(22,42,53,63)(23,48,29,43)(26,46,27,45)(35,59,54,56)(37,58,38,57)(50,61,51,60)(52,64,55,62);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,21)(13,36)(14,39)(15,34)(17,42)(18,43)(19,48)(20,44)(25,54)(26,52)(27,55)(28,53)(31,56)(32,49)(33,41)(37,50)(38,51)(40,59)(45,62)(46,64)(47,63)(57,61)(58,60);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,31)(10,35)(12,37)(13,25)(14,38)(15,40)(16,42)(18,45)(19,46)(20,47)(21,50)(23,52)(24,53)(29,55)(30,56)(32,57)(33,58)(34,59)(36,54)(39,51)(41,60)(43,62)(44,63)(48,64)(49,61);(1,6,8,7)(2,12,13,14)(3,18,20,19)(4,23,24,29)(5,26,28,27)(9,32,15,33)(10,21,36,39)(11,37,25,38)(16,43,44,48)(17,45,47,46)(22,52,53,55)(30,49,34,41)(31,57,40,58)(35,50,54,51)(42,62,63,64)(56,61,59,60)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281403);