Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432197 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#397 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,32,18,33)(4,21,24,35,25,39,29,10)(5,11,28,38,26,36,27,37)(16,42,44,49,22,30,48,40)(17,41,47,31,45,57,46,56)(23,50,53,58,52,59,54,34)(43,61,62,64,51,55,63,60);(1,3,8,20)(2,9,14,33)(4,22,25,16)(5,17,26,45)(6,19,7,18)(10,30,35,42)(11,31,36,56)(12,32,13,15)(21,49,39,40)(23,51,52,43)(24,44,29,48)(27,47,28,46)(34,55,58,61)(37,57,38,41)(50,64,59,60)(53,62,54,63);(1,4,6,24,8,25,7,29)(2,10,12,21,14,35,13,39)(3,16,18,44,20,22,19,48)(5,23,27,53,26,52,28,54)(9,30,15,40,33,42,32,49)(11,34,37,50,36,58,38,59)(17,43,46,62,45,51,47,63)(31,55,41,60,56,61,57,64);(1,5)(2,11)(3,17)(4,23)(6,27)(7,28)(8,26)(9,31)(10,34)(12,37)(13,38)(14,36)(15,41)(16,43)(18,46)(19,47)(20,45)(21,50)(22,51)(24,53)(25,52)(29,54)(30,55)(32,57)(33,56)(35,58)(39,59)(40,60)(42,61)(44,62)(48,63)(49,64);(5,26)(11,36)(17,45)(23,52)(27,28)(31,56)(34,58)(37,38)(41,57)(43,51)(46,47)(50,59)(53,54)(55,61)(60,64)(62,63)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72307021, #72307024, #72307052, … |
| maximal normal subgroup of | #72307021, #72307024, #72307052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339394);