Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,22)(9,20)(10,16)(11,15)(12,17)(13,26)(14,28)(19,24)(21,29)(23,30)(25,33)(27,32)(31,36)(34,35);(5,7,10,20,12,14,11,21)(6,9,15,18,17,29,16,28)(8,23,26,31,27,34,25,19)(13,30,22,24,33,35,32,36);(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,27)(14,31)(15,32)(16,22)(17,33)(18,30)(20,34)(21,23)(28,35)(29,36);(2,4)(5,9)(6,14)(7,17)(8,24)(10,28)(11,18)(12,29)(13,31)(15,20)(16,21)(19,33)(22,23)(25,35)(26,30)(27,36)(32,34)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=dei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328011);