Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13971 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1841 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)(3,15)(4,21,26,36)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25,37,24)(11,28)(16,31,44,49)(17,32)(19,33)(20,40)(22,34,48,41)(23,35,54,60)(29,39)(30,38)(42,45)(43,59,63,61)(46,58)(47,57)(50,53,55,52)(51,64,62,56);(1,3,6,19,8,20,7,18)(2,9,13,15,14,33,12,40)(4,22,24,16,26,48,25,44)(5,17,29,45,30,46,28,47)(10,34,36,31,37,41,21,49)(11,32,27,42,39,58,38,57)(23,51,53,43,54,62,52,63)(35,59,55,56,60,61,50,64);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,35)(12,21)(13,36)(14,37)(15,41)(17,43)(18,22)(19,48)(20,44)(27,55)(28,52)(29,53)(30,54)(32,56)(33,49)(34,40)(38,50)(39,60)(42,61)(45,62)(46,63)(47,51)(57,59)(58,64);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,35)(12,38)(13,27)(14,39)(15,42)(16,43)(18,47)(19,45)(20,46)(21,50)(22,51)(24,53)(25,52)(26,54)(31,56)(33,58)(34,59)(36,55)(37,60)(40,57)(41,61)(44,63)(48,62)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272369);