Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461124 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16380 |
| 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 | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,28,8,12,7,13)(3,14,29,34,17,9,18,33)(4,19,22,37,23,10,30,38)(5,24,25,11,27,40,26,39)(15,41,45,51,44,31,20,35)(16,46,50,42,48,58,49,32)(21,54,55,36,56,59,57,52)(43,62,53,60,47,64,63,61);(1,3)(2,9,12,14)(4,20)(5,16)(6,29)(7,18)(8,17)(10,35,19,51)(11,32,39,42)(13,34,28,33)(15,22)(21,53)(23,45)(24,58,40,46)(25,50)(26,49)(27,48)(30,44)(31,38,41,37)(36,62,52,64)(43,57)(47,55)(54,61,59,60)(56,63);(1,4)(2,10)(3,15)(5,21)(6,30)(7,22)(8,23)(9,31)(11,36)(12,19)(13,38)(14,41)(16,47)(17,44)(18,45)(20,29)(24,59)(25,57)(26,55)(27,56)(28,37)(32,61)(33,51)(34,35)(39,52)(40,54)(42,60)(43,48)(46,64)(49,63)(50,53)(58,62);(1,5,8,27)(2,11,12,39)(3,16,17,48)(4,21,23,56)(6,26,7,25)(9,32,14,42)(10,36,19,52)(13,40,28,24)(15,43,44,47)(18,50,29,49)(20,53,45,63)(22,57,30,55)(31,60,41,61)(33,58,34,46)(35,62,51,64)(37,59,38,54)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368321);