Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422306 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15304 |
| 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 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,18,35,20,34,19,9)(4,21,25,39,26,38,24,10)(5,27,29,11,30,41,31,42)(16,32,45,36,46,59,22,43)(17,47,28,57,49,33,48,40)(23,52,55,61,56,37,54,50)(44,62,63,58,53,60,51,64);(1,3,8,20)(2,9,14,35)(4,22)(5,28,30,48)(6,19,7,18)(10,36)(11,40,42,57)(12,34,13,15)(16,24)(17,31,49,29)(21,32)(23,53)(25,46)(26,45)(27,47,41,33)(37,58)(38,59)(39,43)(44,56)(50,62)(51,54)(52,64)(55,63)(60,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,37)(12,38)(13,21)(14,39)(15,43)(17,44)(18,22)(19,45)(20,46)(27,50)(28,51)(29,54)(30,56)(31,55)(33,58)(34,36)(35,59)(40,62)(41,61)(42,52)(47,64)(48,63)(49,53)(57,60);(1,5,6,29,8,30,7,31)(2,11,12,41,14,42,13,27)(3,17,18,28,20,49,19,48)(4,23,24,54,26,56,25,55)(9,33,15,40,35,47,34,57)(10,37,38,61,39,52,21,50)(16,44,22,51,46,53,45,63)(32,58,43,62,59,64,36,60)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 329502);