Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460666 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16369 |
| 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 | |
899 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)(3,15,19,41)(4,20,24,10)(5,25,28,11)(6,12)(7,14)(8,13)(9,18,34,30)(16,35)(17,44)(21,53)(22,36)(23,42,31,37)(26,38)(27,40,29,39)(32,46)(33,49)(43,47)(45,60,58,63)(48,62,64,59)(50,56)(51,61)(52,55)(54,57);(1,3)(2,9,13,34)(4,21)(5,17)(6,18)(7,30)(8,19)(10,35,20,43)(11,33,25,56)(12,15,14,41)(16,23)(22,48)(24,46)(26,58)(27,49)(28,51)(29,50)(31,47)(32,42,53,37)(36,60,52,63)(38,62,57,59)(39,61,40,44)(45,54)(55,64);(1,4,6,23,8,24,7,31)(2,10,12,37,13,20,14,42)(3,16,18,46,19,47,30,21)(5,26,27,55,28,54,29,22)(9,32,15,43,34,53,41,35)(11,38,39,52,25,57,40,36)(17,48,49,58,51,64,50,45)(33,60,61,62,56,63,44,59);(1,5,8,28)(2,11,13,25)(3,17,19,51)(4,22,24,55)(6,29,7,27)(9,33,34,56)(10,36,20,52)(12,40,14,39)(15,44,41,61)(16,45,47,58)(18,50,30,49)(21,48,46,64)(23,54,31,26)(32,59,53,62)(35,60,43,63)(37,57,42,38)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367863);