Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(3,12)(5,6)(10,21)(11,18)(13,26)(14,19)(16,17)(20,23)(25,28)(27,29)(30,32)(33,34)(35,44)(36,40)(37,49)(38,43)(39,50)(41,47)(42,52)(45,54)(46,61)(48,56)(51,58)(53,57)(55,62)(59,64)(60,63);(1,2)(3,9)(4,10)(5,18)(6,8)(7,11)(12,23)(13,22)(14,24)(15,25)(16,28)(17,21)(19,20)(26,32)(27,31)(29,30)(33,35)(34,41)(36,45)(37,46)(38,47)(39,54)(40,48)(42,55)(43,56)(44,50)(49,60)(51,62)(52,64)(53,63)(57,59)(58,61);(1,3,7,14)(2,9,11,24)(4,13,15,27)(5,12,6,19)(8,20,18,23)(10,22,25,31)(16,26,17,29)(21,30,28,32)(33,36,40,34)(35,45,48,41)(37,49,52,42)(38,50,39,43)(44,54,56,47)(46,60,64,55)(51,58,53,57)(59,62,61,63);(1,4)(2,10)(3,13)(5,16)(6,17)(7,15)(8,21)(9,22)(11,25)(12,26)(14,27)(18,28)(19,29)(20,30)(23,32)(24,31)(33,37)(34,42)(35,46)(36,49)(38,53)(39,51)(40,52)(41,55)(43,58)(44,59)(45,60)(47,63)(48,64)(50,57)(54,62)(56,61);(4,15)(10,25)(13,27)(16,17)(21,28)(22,31)(26,29)(30,32)(33,38,40,39)(34,43,36,50)(35,47,48,54)(37,51,52,53)(41,56,45,44)(42,57,49,58)(46,62,64,63)(55,59,60,61)Polycyclic presentation:
2^9|c^2=h,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259197);