Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187780 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2020 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)(3,11,14,7)(4,8)(5,9)(6,10)(12,27,29,20)(13,24,17,21)(15,22)(16,23)(18,25)(19,26)(28,32,30,31)(33,34)(35,43)(36,47,48,40)(37,50)(38,42)(39,46)(41,52)(44,58,60,57)(45,56)(49,55,63,62)(51,54)(53,59,61,64);(1,3)(2,7)(4,12)(5,17)(6,14)(8,20)(9,24)(10,11)(13,19)(15,30)(16,29)(18,28)(21,26)(22,32)(23,27)(25,31)(33,35)(34,39)(36,44)(37,51)(38,45)(40,53)(41,56)(42,54)(43,50)(46,52)(47,61)(48,60)(49,64)(55,58)(57,62)(59,63);(1,4,6,16)(2,8,10,23)(3,12,14,29)(5,18,19,15)(7,20,11,27)(9,25,26,22)(13,30,17,28)(21,32,24,31)(33,36,38,48)(34,40,42,47)(35,44,45,60)(37,49,52,63)(39,53,54,61)(41,55,50,62)(43,57,56,58)(46,59,51,64);(1,5,6,19)(2,9,10,26)(3,13,14,17)(4,15,16,18)(7,21,11,24)(8,22,23,25)(12,28,29,30)(20,31,27,32)(35,45)(36,48)(39,54)(40,47)(43,56)(46,51)(49,63)(55,62)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=ghi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94994);