Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13,16,23)(4,17)(5,11)(6,10)(7,12)(8,15,24,20)(9,18)(14,28,31,22)(19,25)(21,26)(27,29,32,30)(33,34,38,44,39,42,37,43)(35,40,46,55,47,54,52,58)(36,48,51,41,49,57,50,56)(45,59,61,64,60,63,62,53);(1,3,5,20,7,16,6,15)(2,8,11,23,12,24,10,13)(4,14,18,30,19,31,21,29)(9,22,25,32,26,28,17,27)(33,35,38,46,39,47,37,52)(34,40,44,55,42,54,43,58)(36,45,50,62,49,60,51,61)(41,53,48,63,56,64,57,59);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,22)(10,17)(11,25)(12,26)(13,27)(15,29)(16,31)(20,30)(23,32)(24,28)(33,36)(34,41)(35,45)(37,51)(38,50)(39,49)(40,53)(42,56)(43,57)(44,48)(46,62)(47,60)(52,61)(54,64)(55,63)(58,59);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,18,19,21)(8,23,24,13)(9,17,26,25)(14,29,31,30)(22,32,28,27)(34,42)(40,54)(41,56)(43,44)(48,57)(53,64)(55,58)(59,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101841);