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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12)(3,14,17,37)(4,9,20,33)(5,24,27,10)(7,13,21,11)(8,25,30,16)(15,40,43,64)(18,42,44,41)(19,48,49,61)(22,53,51,34)(23,36,52,35)(26,39,46,38)(28,47,31,57)(29,58,60,45)(32,54,59,50)(55,63,56,62);(1,3,7,18)(2,8,13,31)(4,19,23,15)(5,25,26,57)(6,17,21,44)(9,32,36,29)(10,37,38,41)(11,28,12,30)(14,39,42,24)(16,46,47,27)(20,49,52,43)(22,54,56,58)(33,59,35,60)(34,61,63,64)(40,53,48,62)(45,51,50,55);(4,20)(5,26)(8,28)(9,33)(10,38)(14,37)(15,43)(16,47)(19,49)(22,55)(23,52)(24,39)(25,57)(27,46)(29,32)(30,31)(34,62)(35,36)(41,42)(45,54)(50,58)(51,56)(53,63)(59,60);(1,4,21,52)(2,9,11,35)(3,15,44,49)(5,22,46,55)(6,20,7,23)(8,29,28,59)(10,34,39,62)(12,33,13,36)(14,40,41,61)(16,45,57,54)(17,43,18,19)(24,53,38,63)(25,58,47,50)(26,56,27,51)(30,60,31,32)(37,64,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,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, 142195);