Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2684 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2206 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)(4,21)(5,11)(6,14)(7,13)(8,12)(9,18)(10,24)(16,35)(17,44)(19,33)(20,34)(22,31)(23,53)(25,42)(26,58)(27,41)(28,40)(29,39)(30,37)(32,47)(36,55)(38,48)(43,46)(45,60)(49,50)(51,56)(52,61)(54,64)(57,63)(59,62);(1,3,7,19,8,20,6,18)(2,9,14,34,12,33,13,15)(4,22,24,49,25,16,30,46)(5,26,28,17,29,47,27,48)(10,35,21,50,37,31,42,43)(11,38,41,32,39,44,40,58)(23,54,55,45,56,59,57,52)(36,62,53,60,63,64,51,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31)(11,36)(12,37)(13,21)(14,42)(15,43)(17,45)(18,46)(19,49)(20,22)(26,59)(27,55)(28,57)(29,56)(32,60)(33,35)(34,50)(38,64)(39,63)(40,53)(41,51)(44,62)(47,54)(48,52)(58,61);(1,5,8,29)(2,11,12,39)(3,17,20,48)(4,23,25,56)(6,28,7,27)(9,32,33,58)(10,36,37,63)(13,41,14,40)(15,44,34,38)(16,45,22,52)(18,47,19,26)(21,51,42,53)(24,57,30,55)(31,60,35,61)(43,62,50,64)(46,54,49,59);(2,12)(9,33)(10,37)(11,39)(13,14)(15,34)(21,42)(31,35)(32,58)(36,63)(38,44)(40,41)(43,50)(51,53)(60,61)(62,64)Polycyclic presentation:
2^9|b^2=g,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411489);