Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442123 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15701 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
845 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)(3,15,18,9,20,34,19,35)(4,21,24,39,26,38,25,10)(5,27,31,42,30,11,29,41)(16,32,47,36,44,48,22,50)(17,40,28,33,46,57,45,60)(23,52,54,37,56,49,55,58)(43,59,63,61,53,62,51,64);(1,3,8,18,7,20,6,19)(2,9,12,34,14,35,13,15)(4,22,24,16,26,47,25,44)(5,28,31,46,30,45,29,17)(10,36,21,48,39,50,38,32)(11,40,41,33,27,57,42,60)(23,53,54,51,56,43,55,63)(37,62,49,64,58,59,52,61);(1,4,7,26)(2,10)(3,16,20,44)(5,23,30,56)(6,24,8,25)(9,32)(11,37)(12,38)(13,21)(14,39)(15,36)(17,43,46,53)(18,22,19,47)(27,58)(28,51,45,63)(29,54,31,55)(33,59)(34,50)(35,48)(40,61)(41,52)(42,49)(57,64)(60,62);(1,5,6,29,7,30,8,31)(2,11,12,41,14,27,13,42)(3,17,18,28,20,46,19,45)(4,23,24,54,26,56,25,55)(9,33,15,40,35,60,34,57)(10,37,38,52,39,58,21,49)(16,43,22,51,44,53,47,63)(32,59,36,61,48,62,50,64)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,c^2=ef,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,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 349319);