Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1893 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1893 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,8,13,7,30)(3,14,16,42,19,44,18,43)(4,20,23,39,24,37,21,10)(6,26,25,32,29,57,28,12)(9,31,34,52,36,51,35,17)(15,45,48,41,49,63,46,60)(22,38,56,54,27,40,55,62)(33,47,53,58,61,50,59,64);(1,3,6,17,5,16,25,31,8,19,29,52,7,18,28,51)(2,9,32,44,30,35,26,42,13,36,12,14,11,34,57,43)(4,15,27,50,21,46,56,58,24,49,22,47,23,48,55,64)(10,33,38,60,20,53,54,45,39,61,40,41,37,59,62,63);(1,4)(2,10)(3,15)(5,21)(6,27)(7,23)(8,24)(9,33)(11,37)(12,40)(13,39)(14,41)(16,46)(17,50)(18,48)(19,49)(20,30)(22,29)(25,56)(26,54)(28,55)(31,58)(32,38)(34,59)(35,53)(36,61)(42,45)(43,63)(44,60)(47,52)(51,64)(57,62);(1,5,8,7)(2,11,13,30)(3,16,19,18)(4,21,24,23)(6,25,29,28)(9,34,36,35)(10,37,39,20)(12,26,32,57)(14,42,44,43)(15,46,49,48)(17,31,52,51)(22,55,27,56)(33,59,61,53)(38,62,40,54)(41,45,60,63)(47,64,50,58)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285581, #72285749, #72285775, … |
| maximal normal subgroup of | #72285581, #72285749, #72285775, … |
| maximal quotient of | #72310120, #72321454, #72322852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401849);