Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1964 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C2 | #10854 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,13)(3,15,19,44)(4,21,24,53)(5,26,28,12)(6,30,29,11)(8,14,32,42)(9,33,36,17)(10,38,40,23)(16,46,49,34)(18,51,27,37)(20,45,31,35)(22,55,57,41)(25,54,59,39)(43,56,62,48)(47,52,60,63)(50,64,58,61);(1,3)(2,9,14,37)(4,16)(5,27,29,17)(6,18)(7,31,32,19)(8,20)(10,34,41,61)(11,15)(12,35,30,44)(13,36)(21,52,54,62)(22,56,38,47)(23,48)(24,58,59,49)(25,50)(26,45)(28,33)(39,43)(40,46)(42,51)(53,63)(55,64)(57,60);(1,4,6,23,8,25,28,57)(2,10,12,21,14,41,30,54)(3,16,18,48,20,50,33,60)(5,22,7,24,29,38,32,59)(9,34,35,52,37,61,44,62)(11,39,13,40,26,53,42,55)(15,43,36,46,45,63,51,64)(17,47,19,49,27,56,31,58)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,f^d=fi,g^d=gi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280848, #72281046, #72281176, … |
| maximal normal subgroup of | #72280848, #72281046, #72281176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54272);