Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1169 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)(4,8)(5,17)(6,10)(7,13)(9,19)(12,27)(14,25)(15,23)(16,31)(18,21)(20,32)(22,29)(24,30)(26,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,50)(44,51)(45,52)(46,54)(47,55)(48,56)(49,57)(53,59)(58,62)(60,63)(61,64);(1,3)(2,7)(4,12)(5,18)(6,14)(8,22)(9,25)(10,21)(11,17)(13,19)(15,28)(16,32)(20,23)(24,26)(27,31)(29,30)(33,35,38,45)(34,39,42,52)(36,43,47,59)(37,48,49,44)(40,53,55,50)(41,56,57,51)(46,60,61,58)(54,62,64,63);(1,4,6,15)(2,8,10,23)(3,12,14,28)(5,16,19,30)(7,20,21,22)(9,24,17,31)(11,26,25,27)(13,29,18,32)(33,36)(34,40)(35,43)(37,46)(38,47)(39,50)(41,54)(42,55)(44,58)(45,59)(48,60)(49,61)(51,62)(52,53)(56,63)(57,64);(4,15)(8,23)(12,28)(16,30)(20,22)(24,31)(26,27)(29,32)(33,37,38,49)(34,41,42,57)(35,44,45,48)(36,46,47,61)(39,51,52,56)(40,54,55,64)(43,58,59,60)(50,62,53,63)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142193);