Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451053 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15984 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,19,34,30,35)(4,20,23,10,25,38,24,39)(5,26,28,41,29,11,31,42)(16,44,21,36,47,32,46,43)(17,40,58,60,48,55,27,33)(22,51,53,37,54,49,52,56)(45,63,57,62,64,59,50,61);(1,3)(2,9)(4,21)(5,27)(6,30)(7,19)(8,18)(10,36)(11,40)(12,34)(13,15)(14,35)(16,24)(17,28)(20,44)(22,45)(23,47)(25,46)(26,55)(29,58)(31,48)(32,38)(33,41)(37,59)(39,43)(42,60)(49,61)(50,52)(51,62)(53,57)(54,64)(56,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,25)(8,24)(9,32)(11,37)(12,20)(13,38)(14,39)(15,43)(17,45)(18,46)(19,47)(21,30)(26,56)(27,57)(28,52)(29,54)(31,53)(33,59)(34,36)(35,44)(40,62)(41,49)(42,51)(48,64)(50,58)(55,61)(60,63);(1,5,6,28,7,29,8,31)(2,11,12,41,14,26,13,42)(3,17,18,27,19,48,30,58)(4,22,23,52,25,54,24,53)(9,33,15,40,35,60,34,55)(10,37,20,49,39,56,38,51)(16,45,46,57,47,64,21,50)(32,59,43,62,44,63,36,61)Polycyclic presentation:
2^9|a^2=gh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358249);