Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 | C22×C4 | C25 |
| prime radical | #62964 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,12)(4,9)(5,17)(6,20)(7,11)(8,13)(10,19)(14,29)(15,22)(16,26)(18,25)(21,24)(23,30)(27,31)(28,32)(33,34,39,38)(35,40,46,45)(36,41,49,48)(37,42,50,51)(43,55,57,52)(44,53,58,59)(47,54,60,61)(56,63,64,62);(1,3)(2,8)(4,13)(5,14)(6,21)(7,15)(9,22)(10,23)(11,24)(12,20)(16,28)(17,32)(18,30)(19,31)(25,29)(26,27)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,52)(42,53)(47,56)(48,55)(49,57)(50,58)(51,59)(54,62)(60,64)(61,63);(1,4,7,6)(2,9,11,20)(3,13,15,21)(5,16,19,18)(8,22,24,12)(10,25,17,26)(14,28,31,30)(23,29,32,27)(33,36)(34,41)(35,43)(37,47)(38,48)(39,49)(40,52)(42,54)(44,56)(45,55)(46,57)(50,60)(51,61)(53,62)(58,64)(59,63);(1,5)(2,10)(3,14)(4,16)(6,18)(7,19)(8,23)(9,25)(11,17)(12,27)(13,28)(15,31)(20,26)(21,30)(22,29)(24,32)(33,37,39,50)(34,42,38,51)(35,44,46,58)(36,47,49,60)(40,53,45,59)(41,54,48,61)(43,56,57,64)(52,62,55,63);(1,4,7,6)(2,9,11,20)(3,13,15,21)(5,18,19,16)(8,22,24,12)(10,26,17,25)(14,30,31,28)(23,27,32,29)(33,38,39,34)(35,45,46,40)(36,48,49,41)(37,42,50,51)(43,55,57,52)(44,53,58,59)(47,54,60,61)(56,62,64,63)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260405);