Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343678 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62838 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,5,10,6,9,18,23)(3,11,13,24,14,7,17,20)(4,8,16,22,15,21,28,31)(12,25,27,32,26,19,29,30)(33,34)(35,44,47,54)(36,41)(37,51)(38,42)(39,43)(40,46,55,52)(45,58,59,64)(48,56)(49,61)(50,57)(53,60,63,62);(1,3,6,14)(2,7,9,11)(4,12,15,26)(5,17,18,13)(8,19,21,25)(10,24,23,20)(16,29,28,27)(22,32,31,30)(33,35)(34,40)(36,45)(37,46)(38,52)(39,47)(41,53)(42,54)(43,55)(44,51)(48,59)(49,60)(50,62)(56,63)(57,64)(58,61);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,21)(10,22)(11,25)(13,27)(14,26)(17,29)(18,28)(20,30)(23,31)(24,32)(33,36)(34,41)(35,45)(37,49)(38,50)(39,48)(40,53)(42,57)(43,56)(44,58)(46,60)(47,59)(51,61)(52,62)(54,64)(55,63);(36,48)(41,56)(45,59)(49,50)(53,63)(57,61)(58,64)(60,62);(4,15)(8,21)(12,26)(16,28)(19,25)(22,31)(27,29)(30,32)(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,49,48,50)(40,54,55,44)(41,57,56,61)(45,60,59,62)(53,64,63,58)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=i,g^2=h,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251151);