Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| 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 | #62879 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,11,14,24)(4,8,15,22)(5,17,19,10)(7,18,21,13)(12,25,26,32)(16,29,28,23)(20,30,31,27)(39,50)(51,61)(52,62)(63,64);(1,3,6,14)(2,7,9,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(10,24,17,11)(16,30,28,27)(23,32,29,25)(33,34)(35,39)(36,40)(37,41)(38,42)(43,51)(44,52)(45,50)(46,53)(47,54)(48,55)(49,56)(57,63)(58,61)(59,64)(60,62);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,22)(10,23)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,51)(41,52)(42,50)(46,57)(47,58)(48,59)(49,60)(53,63)(54,61)(55,64)(56,62);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,36,38,47)(34,40,42,54)(35,43,45,58)(37,48,49,46)(39,51,50,61)(41,55,56,53)(44,59,60,57)(52,64,62,63);(2,9)(7,21)(8,22)(10,17)(11,24)(20,31)(23,29)(25,32)(33,37,38,49)(34,41,42,56)(35,44,45,60)(36,46,47,48)(39,52,50,62)(40,53,54,55)(43,57,58,59)(51,63,61,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259714);