Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792410 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #38904 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1798 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)(3,14)(4,20)(5,7)(6,12)(8,28)(9,13)(10,17)(11,24)(15,44)(16,43)(18,35)(19,40)(21,33)(22,25)(23,50)(26,41)(27,30)(29,57)(31,58)(32,39)(34,48)(36,47)(37,54)(38,49)(42,46)(45,63)(51,59)(52,55)(53,60)(56,62)(61,64);(1,3,7,17)(2,10,5,14)(4,21,26,46)(6,29,30,49)(8,19,9,18)(11,36,22,44)(12,38,27,57)(13,40,28,35)(15,24,47,25)(16,32,48,31)(20,33,41,42)(23,53,56,45)(34,39,43,58)(37,61,52,59)(50,60,62,63)(51,54,64,55);(1,4)(2,11)(3,15)(5,22)(6,23)(7,26)(8,25)(9,24)(10,33)(12,37)(13,41)(14,42)(16,45)(17,47)(18,21)(19,46)(20,28)(27,52)(29,51)(30,56)(31,55)(32,54)(34,59)(35,44)(36,40)(38,60)(39,62)(43,61)(48,53)(49,64)(50,58)(57,63);(1,5,7,2)(3,14,17,10)(4,22,26,11)(6,27,30,12)(8,13,9,28)(15,42,47,33)(16,43,48,34)(18,35,19,40)(20,25,41,24)(21,44,46,36)(23,52,56,37)(29,57,49,38)(31,39,32,58)(45,61,53,59)(50,55,62,54)(51,63,64,60);(1,6,9,32,7,30,8,31)(2,12,13,39,5,27,28,58)(3,16,19,49,17,48,18,29)(4,23,24,54,26,56,25,55)(10,34,35,57,14,43,40,38)(11,37,41,62,22,52,20,50)(15,45,46,64,47,53,21,51)(33,59,44,63,42,61,36,60)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699611);