Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13960 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1373 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)(3,15,19,38)(4,20,24,39)(5,25,28,36)(6,12)(7,13)(8,14)(9,18,33,29)(10,23,35,30)(11,26,37,27)(16,31)(17,45)(21,49)(22,51)(32,47)(34,53)(40,43)(41,54)(42,60,50,64)(44,57)(46,58)(48,55)(52,59)(56,61,63,62);(1,3,7,29,8,19,6,18)(2,9,12,15,14,33,13,38)(4,21,30,43,24,16,23,44)(5,17,27,54,28,47,26,46)(10,31,20,40,35,49,39,57)(11,32,25,41,37,45,36,58)(22,50,55,61,53,42,52,62)(34,56,48,60,51,63,59,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,34)(12,20)(13,39)(14,35)(15,40)(17,42)(18,43)(19,21)(25,48)(26,52)(27,55)(28,53)(29,44)(32,56)(33,49)(36,59)(37,51)(38,57)(41,60)(45,63)(46,61)(47,50)(54,62)(58,64);(1,5,7,27,8,28,6,26)(2,11,12,25,14,37,13,36)(3,17,29,54,19,47,18,46)(4,22,30,55,24,53,23,52)(9,32,15,41,33,45,38,58)(10,34,20,48,35,51,39,59)(16,42,44,62,21,50,43,61)(31,56,40,60,49,63,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285784, #72285922, #72286119, … |
| maximal normal subgroup of | #72285784, #72285922, #72286119, … |
| maximal quotient of | C2×#364128 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271324);