Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #430079 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15412 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1494 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)(3,8,15,29)(4,9)(5,21)(6,24)(7,11)(10,23)(12,26)(13,31,17,27)(14,40,41,54)(16,44,43,30)(18,48)(19,34)(20,52)(22,37)(25,36)(28,45,35,42)(32,50)(33,51)(38,61,46,64)(39,56,47,58)(49,60)(53,59)(55,62,57,63);(1,3)(2,8,11,29)(4,17,19,13)(5,14,22,41)(6,16)(7,15)(9,31)(10,35)(12,30,24,44)(18,38)(20,47,51,39)(21,54)(23,42,25,45)(26,43)(27,34)(28,36)(32,57,59,55)(33,58)(37,40)(46,49)(48,61,60,64)(50,62)(52,56)(53,63);(1,4,7,19)(2,9,11,34)(3,13,15,17)(5,18,22,49)(6,20,26,51)(8,27,29,31)(10,32,36,59)(12,33,24,52)(14,38,41,46)(16,39,43,47)(21,48,37,60)(23,50,25,53)(28,55,35,57)(30,56,44,58)(40,64,54,61)(42,62,45,63);(1,5)(2,10)(3,14,15,41)(4,18)(6,25)(7,22)(8,28,29,35)(9,32)(11,36)(12,37)(13,38,17,46)(16,45,43,42)(19,49)(20,53)(21,24)(23,26)(27,55,31,57)(30,54,44,40)(33,60)(34,59)(39,63,47,62)(48,52)(50,51)(56,61,58,64)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72283686, #72319969, #72319988, … |
| maximal normal subgroup of | #72283686, #72319969, #72319988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337276);