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 | #62869 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,26)(4,9,17,23)(5,11,6,10)(8,20,22,15)(14,27,28,32)(18,25,19,24)(21,30,31,29)(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)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,29)(19,30)(23,31)(24,27)(25,32)(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)(2,9)(3,14)(5,18)(6,19)(7,17)(8,21)(10,24)(11,25)(12,23)(13,27)(15,29)(16,28)(20,30)(22,31)(26,32)(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);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(29,30)(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);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,18,17,19)(8,13,22,26)(9,24,23,25)(14,29,28,30)(21,27,31,32)(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=hi,d^2=i,e^2=hi,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, 6259773);