Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2077 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11378 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,12,18,38,33,13,9,16,5,32,8,15,6,14)(3,17,20,43,36,41,35,10,23,42,34,51,22,40,21,44)(4,24,29,48,37,50,28,47,31,49,26,11,30,46,27,52)(19,53,54,63,58,39,25,60,57,64,61,45,56,59,55,62);(1,3,18,22,9,23,8,36)(2,10,38,43,16,44,15,51)(4,25,37,54,31,55,30,61)(5,21,6,34,7,35,33,20)(11,45,52,60,48,63,47,62)(12,40,13,42,32,41,14,17)(19,29,56,28,57,26,58,27)(24,59,50,64,49,39,46,53);(1,4,9,31)(2,11,16,48)(3,19)(5,28,7,27)(6,29,33,26)(8,37,18,30)(10,39)(12,49,32,24)(13,50,14,46)(15,52,38,47)(17,45)(20,55)(21,54)(22,58)(23,57)(25,34)(35,61)(36,56)(40,62)(41,60)(42,63)(43,64)(44,59)(51,53)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=h,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56594);