Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395638 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14725 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2075 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)(3,14)(4,10)(5,25,27,36)(6,28)(7,12)(8,13)(9,17)(11,26,37,29)(15,39)(16,31,46,58)(18,41)(19,32)(20,48)(21,50,52,55)(22,38)(23,34)(24,35)(30,44)(33,51,59,53)(40,45,54,47)(42,61,49,56)(43,57)(60,62,64,63);(1,3,6,17,8,18,7,19)(2,9,12,14,13,32,28,41)(4,20,22,44,24,15,23,43)(5,16,26,45,27,46,29,47)(10,30,34,48,35,57,38,39)(11,31,36,40,37,58,25,54)(21,49,51,63,52,42,53,62)(33,56,55,64,59,61,50,60);(1,4,8,24)(2,10,13,35)(3,15,18,20)(5,21,27,52)(6,23,7,22)(9,30,32,57)(11,33,37,59)(12,38,28,34)(14,39,41,48)(16,42,46,49)(17,44,19,43)(25,55,36,50)(26,53,29,51)(31,56,58,61)(40,60,54,64)(45,63,47,62);(1,5)(2,11)(3,16)(4,21)(6,26)(7,29)(8,27)(9,31)(10,33)(12,36)(13,37)(14,40)(15,42)(17,45)(18,46)(19,47)(20,49)(22,51)(23,53)(24,52)(25,28)(30,56)(32,58)(34,55)(35,59)(38,50)(39,60)(41,54)(43,62)(44,63)(48,64)(57,61)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302835);