Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200146 |
| 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 | #60959 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
977 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)(4,15,17,8)(5,18)(6,10)(7,14)(9,20)(12,25,27,32)(13,26)(16,23,19,24)(21,28,31,30)(22,29)(33,34)(35,39,44,54)(36,46,47,58)(37,42)(38,41)(40,48,55,50)(43,59)(45,53,49,57)(51,56)(52,62)(60,63)(61,64);(1,3,5,14,6,13,20,29)(2,7,9,11,10,22,18,26)(4,12,19,30,17,27,16,28)(8,21,24,25,15,31,23,32)(33,35)(34,39,41,54)(36,43)(37,49)(38,44)(40,52,55,64)(42,57,56,53)(45,51)(46,63,58,59)(47,60)(48,62)(50,61);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,15)(11,25)(13,27)(14,30)(16,20)(18,23)(22,31)(26,32)(28,29)(33,36)(34,40)(35,43)(37,50)(38,47)(39,52)(41,55)(42,58)(44,60)(45,62)(46,56)(48,51)(49,61)(53,63)(54,64)(57,59);(3,13)(7,22)(11,26)(12,27)(14,29)(21,31)(25,32)(28,30)(34,41)(35,44)(40,55)(42,56)(43,60)(45,49)(46,58)(61,62)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107342);