Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1879 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1879 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,30,34,32,12,8,14,29,47,5,11,6,31)(3,15,19,9,17,37,18,36,21,49,55,39,20,45,33,38)(4,22,26,44,23,46,24,43,28,41,59,10,27,42,25,40)(16,35,53,48,57,64,52,61,54,63,56,58,50,60,51,62);(1,3,5,17,8,21,30,20)(2,9,34,38,14,39,11,36)(4,16,23,50,28,54,27,57)(6,19,29,18,32,55,7,33)(10,35,43,64,44,63,40,60)(12,45,47,49,31,37,13,15)(22,58,42,62,41,48,46,61)(24,56,59,52,25,53,26,51);(1,4)(2,10)(3,16)(5,23)(6,26)(7,25)(8,28)(9,35)(11,40)(12,46)(13,41)(14,44)(15,48)(17,50)(18,56)(19,51)(20,57)(21,54)(22,47)(24,29)(27,30)(31,42)(32,59)(33,53)(34,43)(36,60)(37,62)(38,64)(39,63)(45,61)(49,58)(52,55);(1,5,8,30)(2,11,14,34)(3,17,21,20)(4,23,28,27)(6,29,32,7)(9,36,39,38)(10,40,44,43)(12,13,31,47)(15,37,49,45)(16,50,54,57)(18,55,33,19)(22,46,41,42)(24,59,25,26)(35,60,63,64)(48,62,58,61)(51,56,52,53)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285805, #72285921, #72286016, … |
| maximal normal subgroup of | #72285805, #72285921, #72286016, … |
| maximal quotient of | #72321697, #72322927, #72342617 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401746);