Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394366 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2525 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)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,28)(16,31)(17,46)(19,34)(20,33)(22,35)(23,36)(24,38)(26,37)(29,39)(30,40)(32,49)(41,44)(42,47)(43,61)(45,50)(48,58)(51,53)(52,57)(54,56)(55,60)(59,63)(62,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,45,26,44,25,16)(5,17,28,47,29,49,30,48)(10,35,37,31,38,41,21,50)(11,32,27,42,40,46,39,58)(23,52,53,63,55,62,54,43)(36,59,56,57,60,61,51,64);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,37)(13,21)(14,38)(15,41)(17,43)(18,22)(19,44)(20,45)(27,56)(28,53)(29,55)(30,54)(32,57)(33,35)(34,50)(39,51)(40,60)(42,61)(46,64)(47,52)(48,62)(49,63)(58,59);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,39)(14,40)(15,42)(16,43)(18,47)(19,48)(20,49)(21,51)(22,52)(24,53)(25,54)(26,55)(31,57)(33,58)(34,46)(35,59)(37,56)(38,60)(41,61)(44,62)(45,63)(50,64)Polycyclic presentation:
2^9|b^2=e,e^2=i,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,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72264906, #72264928, #72264996, … |
| maximal normal subgroup of | #72264906, #72264928, #72264996, … |
| maximal quotient of | #72265234 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301562);