Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201978 |
| 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 | |
946 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)(4,18)(5,10)(6,11)(7,12)(8,16)(9,20)(14,28)(15,24)(17,23)(19,25)(21,26)(22,30)(27,29)(31,32)(33,34,37,42,38,41,50,55)(35,39,45,53,44,52,59,63)(36,46,48,56,47,40,49,54)(43,51,58,62,57,61,60,64);(1,3,5,16,7,17,6,15)(2,8,11,13,12,24,10,23)(4,14,20,29,21,30,19,31)(9,22,18,27,25,28,26,32)(33,35,37,45,38,44,50,59)(34,39,42,53,41,52,55,63)(36,43,49,60,47,57,48,58)(40,51,56,64,46,61,54,62);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,24,28)(10,26,11,18)(13,27,23,32)(15,31,16,29)(33,36)(34,40)(35,43)(37,49)(38,47)(39,51)(41,46)(42,56)(44,57)(45,60)(48,50)(52,61)(53,64)(54,55)(58,59)(62,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,18,25,26)(14,29,30,31)(22,32,28,27)(34,41)(39,52)(40,46)(42,55)(51,61)(53,63)(54,56)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109322);