Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23185 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,15,8,14,26,47,9,16,25,11,6,32,4,13)(3,17,22,39,21,46,20,36,23,34,18,10,33,37,49,38)(5,12,29,44,30,43,54,64,31,45,53,40,28,58,24,42)(19,41,51,62,50,63,27,55,52,59,48,35,56,60,57,61);(1,3,8,33,9,23,6,21)(2,10,14,36,16,39,32,38)(4,18,7,20,26,22,25,49)(5,27,30,51,31,57,28,48)(11,34,13,46,15,17,47,37)(12,41,43,60,45,59,58,63)(19,29,56,54,52,53,50,24)(35,44,55,64,62,40,61,42);(1,4,6,25,9,26,8,7)(2,11,14,13,16,15,32,47)(3,18,21,49,23,22,33,20)(5,24,28,53,31,54,30,29)(10,34,36,46,39,17,38,37)(12,40,43,42,45,44,58,64)(19,48,50,57,52,51,56,27)(35,59,55,63,62,41,61,60);(1,5,7,29,8,30,26,54,9,31,25,53,6,28,4,24)(2,12,15,44,14,43,47,64,16,45,11,40,32,58,13,42)(3,19,18,48,21,50,49,57,23,52,22,51,33,56,20,27)(10,35,46,63,38,61,34,59,39,62,37,60,36,55,17,41)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=eh,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72225388, #72225403, #72225406, … |
| maximal normal subgroup of | #72225388, #72225403, #72225406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405765);