Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449117 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15931 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,19,33,17,34,18,41)(4,20,23,10,24,38,30,37)(5,25,28,39,29,40,27,11)(15,35,21,48,43,60,44,31)(16,45,46,61,26,56,47,32)(22,51,54,49,55,36,53,57)(42,59,50,62,58,63,52,64);(1,3,6,18,8,17,7,19)(2,9,13,33,12,34,14,41)(4,21,23,15,24,44,30,43)(5,26,27,47,29,16,28,46)(10,35,20,48,37,60,38,31)(11,32,25,56,39,61,40,45)(22,52,53,42,55,50,54,58)(36,62,57,59,51,64,49,63);(1,4)(2,10,12,37)(3,15,17,43)(5,22)(6,23)(7,30)(8,24)(9,31)(11,36,39,51)(13,20,14,38)(16,42,26,58)(18,44,19,21)(25,57,40,49)(27,53)(28,54)(29,55)(32,59)(33,35)(34,48)(41,60)(45,62)(46,50,47,52)(56,64)(61,63);(1,5,6,28,8,29,7,27)(2,11,13,40,12,39,14,25)(3,16,18,47,17,26,19,46)(4,22,23,54,24,55,30,53)(9,32,33,45,34,61,41,56)(10,36,20,49,37,51,38,57)(15,42,44,52,43,58,21,50)(31,59,35,62,48,63,60,64)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356313);