Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,8,18,21)(9,13,22,30)(10,15)(11,14)(12,20,25,26)(16,24)(17,19)(23,29,27,31)(28,36)(32,35)(33,34);(2,4)(5,7,11,25,10,18,24,12)(6,8,14,26,15,21,16,20)(9,17,27,33,22,32,23,28)(13,19,31,34,30,35,29,36);(1,2,3,4)(5,8,10,21)(6,12,15,25)(7,16,18,14)(9,19,22,35)(11,26,24,20)(13,28,30,33)(17,29,32,31)(23,36,27,34);(5,9)(6,13)(7,17)(8,19)(10,22)(11,27)(12,28)(14,31)(15,30)(16,29)(18,32)(20,36)(21,35)(23,24)(25,33)(26,34)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275664);