Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62762 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2686 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,27)(16,22)(17,24)(18,26)(19,25)(20,23)(21,28)(29,32)(30,31)(33,34,37,42,38,41,48,54)(35,39,45,50,44,51,57,61)(36,40,47,53,46,52,58,62)(43,49,56,60,55,59,63,64);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,22)(11,13)(12,23)(17,28)(18,29)(19,30)(24,31)(25,32)(26,27)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,51)(42,50)(46,55)(47,56)(48,57)(52,59)(53,60)(54,61)(58,63)(62,64);(1,4)(2,9)(3,14)(5,17)(6,18)(7,19)(8,21)(10,24)(11,26)(12,25)(13,27)(15,28)(16,30)(20,29)(22,31)(23,32)(33,36)(34,40)(35,43)(37,47)(38,46)(39,49)(41,52)(42,53)(44,55)(45,56)(48,58)(50,60)(51,59)(54,62)(57,63)(61,64);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,17,19,18)(8,22,23,13)(9,24,25,26)(14,28,30,29)(21,31,32,27)(36,46)(40,52)(43,55)(47,58)(49,59)(53,62)(56,63)(60,64);(1,5,7,6)(2,11,12,10)(3,15,16,20)(4,18,19,17)(8,13,23,22)(9,24,25,26)(14,29,30,28)(21,31,32,27)(34,41)(35,44)(40,52)(42,54)(43,55)(45,57)(53,62)(56,63)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,e^b=ei,f^b=fh,d^c=di,e^c=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250471);