Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 ◁ #2920 ◁ 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)(3,13)(4,9)(5,12)(6,10)(7,11)(8,15)(14,27)(16,26)(17,24)(18,25)(19,23)(20,22)(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)(10,13)(11,22)(12,26)(17,28)(18,29)(19,30)(23,27)(24,32)(25,31)(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);(1,4,7,18)(2,9,11,25)(3,14,16,29)(5,17,6,19)(8,21,22,31)(10,23,12,24)(13,27,26,32)(15,28,20,30)(33,36,37,47,38,46,48,58)(34,40,42,53,41,52,54,62)(35,43,45,56,44,55,57,63)(39,49,50,60,51,59,61,64);(1,5,7,6)(2,10,11,12)(3,15,16,20)(4,17,18,19)(8,13,22,26)(9,23,25,24)(14,28,29,30)(21,27,31,32)(36,46)(40,52)(43,55)(47,58)(49,59)(53,62)(56,63)(60,64);(2,11)(4,18)(8,22)(10,12)(13,26)(14,29)(17,19)(28,30)(34,41)(35,44)(40,52)(42,54)(43,55)(45,57)(53,62)(56,63)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,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, 6252866);