Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
970 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)(3,11)(4,15,16,8)(5,18)(6,10)(7,13)(9,20)(12,25,27,32)(14,26)(17,23,19,24)(21,28,31,29)(22,30)(33,34)(35,40,47,55)(36,48,50,56)(37,42)(38,44)(39,43)(41,51,57,49)(45,59)(46,54,52,58)(53,62)(60,64)(61,63);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,18,26)(4,12,19,29,16,27,17,28)(8,21,24,25,15,31,23,32)(33,35)(34,40,43,55)(36,45)(37,46)(38,52)(39,47)(41,53,57,64)(42,54,44,58)(48,63,56,59)(49,60)(50,61)(51,62);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,24)(10,15)(11,25)(13,29)(14,27)(17,20)(18,23)(22,31)(26,32)(28,30)(33,36)(34,41)(35,45)(37,51)(38,49)(39,50)(40,53)(42,48)(43,57)(44,56)(46,62)(47,61)(52,60)(54,63)(55,64)(58,59);(4,16)(8,15)(12,27)(17,19)(21,31)(23,24)(25,32)(28,29)(33,37,39,38)(34,42,43,44)(35,46,47,52)(36,49,50,51)(40,54,55,58)(41,56,57,48)(45,60,61,62)(53,59,64,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107327);