Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199522 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,12,5,11)(3,13,17,23,15,26,16,8)(4,18,20,9,21,24,19,25)(14,22,30,32,28,31,29,27)(33,34)(35,44,48,55)(36,49)(37,42)(38,52)(39,43)(40,47,56,46)(41,50)(45,60,63,54)(51,57)(53,58)(59,62,64,61);(1,3,7,15)(2,8)(4,14,21,28)(5,16,6,17)(9,22)(10,13)(11,26)(12,23)(18,27)(19,29,20,30)(24,32)(25,31)(33,35,37,46,39,48,38,47)(34,40,42,55,43,56,52,44)(36,45,50,61,51,63,53,62)(41,54,57,64,58,60,49,59);(1,4,7,21)(2,9,12,25)(3,14,15,28)(5,20,6,19)(8,22,23,31)(10,18,11,24)(13,27,26,32)(16,30,17,29)(33,36)(34,41)(35,45)(37,50)(38,53)(39,51)(40,54)(42,57)(43,58)(44,59)(46,61)(47,62)(48,63)(49,52)(55,64)(56,60);(3,15)(8,23)(13,26)(14,28)(16,17)(22,31)(27,32)(29,30)(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,50,51,53)(40,55,56,44)(41,57,58,49)(45,61,63,62)(54,64,60,59)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106776);