Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1488 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)(5,11)(6,19)(7,10)(8,14)(13,27)(15,23)(16,26)(17,24)(18,25)(20,22)(21,30)(28,32)(29,31)(33,34,37,42,38,41,48,54)(35,39,45,51,44,50,57,61)(36,40,46,53,47,52,58,62)(43,49,56,60,55,59,63,64);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,28)(17,29)(18,30)(24,27)(25,32)(26,31)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,50)(42,51)(46,56)(47,55)(48,57)(52,59)(53,60)(54,61)(58,63)(62,64);(1,4,7,16)(2,9,10,26)(3,13,15,28)(5,18,6,17)(8,21,22,31)(11,25,19,24)(12,27,23,32)(14,30,20,29)(33,36,37,46,38,47,48,58)(34,40,42,53,41,52,54,62)(35,43,45,56,44,55,57,63)(39,49,51,60,50,59,61,64);(2,10)(4,16)(8,22)(11,19)(12,23)(13,28)(17,18)(29,30)(35,44)(39,50)(43,55)(45,57)(49,59)(51,61)(56,63)(60,64);(1,5,7,6)(2,11,10,19)(3,14,15,20)(4,17,16,18)(8,23,22,12)(9,24,26,25)(13,29,28,30)(21,27,31,32)(34,41)(39,50)(40,52)(42,54)(49,59)(51,61)(53,62)(60,64)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,d^b=di,f^b=fh,d^c=dh,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252445);