Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2644 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2644 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1149 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)(5,27)(6,12)(7,13)(8,14)(9,19)(10,25)(11,29)(16,34)(17,46)(18,39)(20,33)(22,31)(23,52)(24,36)(26,40)(28,38)(30,37)(32,47)(35,55)(41,44)(42,49)(43,61)(45,58)(48,56)(50,54)(51,59)(53,60)(57,63)(62,64);(1,3,6,18,8,19,7,20)(2,9,13,39,14,15,12,33)(4,22,25,45,26,44,24,16)(5,17,28,47,30,48,29,49)(10,34,21,41,36,58,40,31)(11,32,37,46,38,42,27,56)(23,51,54,63,55,62,53,43)(35,59,50,61,52,64,60,57);(1,4)(2,10)(3,16,19,45)(5,23)(6,24)(7,25)(8,26)(9,31,15,41)(11,35)(12,21)(13,40)(14,36)(17,43,48,63)(18,44,20,22)(27,50)(28,53)(29,54)(30,55)(32,57,42,61)(33,34,39,58)(37,60)(38,52)(46,64,56,59)(47,62,49,51);(1,5,7,29,8,30,6,28)(2,11,12,27,14,38,13,37)(3,17,20,49,19,48,18,47)(4,23,25,54,26,55,24,53)(9,32,33,56,15,42,39,46)(10,35,21,50,36,52,40,60)(16,43,22,51,45,63,44,62)(31,57,34,59,41,61,58,64);(1,6,8,7)(2,12,14,13)(3,18,19,20)(4,24,26,25)(5,28,30,29)(9,33,15,39)(10,21,36,40)(11,27,38,37)(16,44,45,22)(17,47,48,49)(23,53,55,54)(31,34,41,58)(32,56,42,46)(35,50,52,60)(43,62,63,51)(57,59,61,64)Polycyclic presentation:
2^9|b^2=fg,c^2=f,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343085);