Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198135 |
| 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 | |
959 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)(3,13,16,24,17,23,15,8)(4,18,20,9,21,25,19,26)(14,22,29,31,30,32,28,27)(33,34)(35,45,49,56)(36,50,53,58)(37,43)(38,44)(39,42)(40,47,55,48)(41,52,57,51)(46,60)(54,63)(59,61)(62,64);(1,3,7,17)(2,8)(4,14,21,30)(5,15,6,16)(9,22)(10,23)(11,13)(12,24)(18,27)(19,28,20,29)(25,31)(26,32)(33,35,37,48,39,49,38,47)(34,40,43,56,42,55,44,45)(36,46,52,61,53,63,51,62)(41,54,50,64,57,60,58,59);(1,4,7,21)(2,9,12,26)(3,14,17,30)(5,20,6,19)(8,22,24,32)(10,25,11,18)(13,27,23,31)(15,29,16,28)(33,36)(34,41)(35,46)(37,52)(38,51)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,62)(48,61)(49,63)(55,60)(56,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,18,26,25)(14,28,30,29)(22,27,32,31)(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=egh,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,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105407);