Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359528 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,26,35)(3,13,16,36,17,8,24,29)(4,9,20,33,19,32,47,58)(5,21,23,37,22,10,25,34)(14,30,41,56,40,27,48,52)(15,28,43,53,42,38,44,54)(18,31,46,57,45,50,49,59)(39,55,61,64,60,51,62,63);(1,3,7,17)(2,8,11,13)(4,14,19,40)(5,15,22,42)(6,24,26,16)(9,30,32,27)(10,28,21,38)(12,36,35,29)(18,39,45,60)(20,48,47,41)(23,44,25,43)(31,55,50,51)(33,52,58,56)(34,54,37,53)(46,62,49,61)(57,63,59,64);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,27)(10,31)(11,32)(12,33)(13,30)(15,39)(16,41)(17,40)(21,50)(22,45)(23,46)(24,48)(25,49)(26,47)(28,51)(29,52)(34,57)(35,58)(36,56)(37,59)(38,55)(42,60)(43,61)(44,62)(53,63)(54,64);(1,5)(2,10)(3,15)(4,18)(6,25)(7,22)(8,28)(9,31)(11,21)(12,37)(13,38)(14,39)(16,44)(17,42)(19,45)(20,49)(23,26)(24,43)(27,51)(29,54)(30,55)(32,50)(33,59)(34,35)(36,53)(40,60)(41,62)(46,47)(48,61)(52,64)(56,63)(57,58)Polycyclic presentation:
2^9|a^2=h,b^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72303145, #72303161, #72303172, … |
| maximal normal subgroup of | #72303145, #72303161, #72303172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266726);