Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14025 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1781 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,19,33,18,39,17,9)(4,20,25,49,24,10,23,35)(5,26,28,36,29,57,27,11)(15,37,42,47,21,50,41,31)(16,43,46,60,45,32,44,38)(22,52,55,34,54,56,53,48)(40,61,63,64,51,58,62,59);(1,3)(2,9,13,33)(4,21)(5,16)(6,17)(7,19)(8,18)(10,31,20,47)(11,32,36,43)(12,14,30,39)(15,24)(22,51)(23,42)(25,41)(26,38,57,60)(27,44)(28,46)(29,45)(34,58,48,61)(35,37,49,50)(40,54)(52,64,56,59)(53,63)(55,62);(1,4)(2,10,13,20)(3,15,18,21)(5,22)(6,23)(7,25)(8,24)(9,31)(11,34,36,48)(12,35,30,49)(14,37)(16,40,45,51)(17,41,19,42)(26,56,57,52)(27,53)(28,55)(29,54)(32,58)(33,47)(38,59)(39,50)(43,61)(44,62,46,63)(60,64);(1,5,6,27,8,29,7,28)(2,11,12,26,13,36,30,57)(3,16,17,44,18,45,19,46)(4,22,23,53,24,54,25,55)(9,32,14,38,33,43,39,60)(10,34,35,56,20,48,49,52)(15,40,41,62,21,51,42,63)(31,58,37,59,47,61,50,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319946, #72319965, #72319975, … |
| maximal normal subgroup of | #72319946, #72319965, #72319975, … |
| maximal quotient of | #72343106 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276394);