Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199558 |
| 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 | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,5,10)(3,13,15,23,17,26,16,8)(4,18,20,9,21,24,19,25)(14,22,28,31,30,32,29,27)(33,34)(35,44,47,56)(36,48)(37,43)(38,52)(39,42)(40,46,55,51)(41,49)(45,60,63,54)(50,58)(53,57)(59,61,64,62);(1,3)(2,8,12,23)(4,14)(5,16)(6,15)(7,17)(9,22,25,32)(10,26,11,13)(18,27,24,31)(19,29)(20,28)(21,30)(33,35,37,51,39,47,38,46)(34,40,43,56,42,55,52,44)(36,45,49,62,50,63,53,61)(41,54,58,64,57,60,48,59);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,23,32)(10,24,11,18)(13,27,26,31)(15,29,16,28)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,58)(44,59)(46,61)(47,63)(48,52)(51,62)(55,60)(56,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,26)(9,18,25,24)(14,28,30,29)(22,31,32,27)(34,42)(40,55)(41,57)(43,52)(44,56)(48,58)(54,60)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106826);