Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62853 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,18)(7,19,22,32)(8,23,26,34)(9,27,29,31)(10,17,11,16)(13,21,33,20)(14,24,35,25)(15,30,36,28);(1,2)(3,4)(5,7)(6,13)(8,9)(10,21)(11,20)(12,22)(14,31)(15,23)(16,32)(17,19)(18,33)(24,30)(25,28)(26,29)(27,35)(34,36);(2,4)(5,8)(6,14)(7,9)(10,24)(11,25)(12,26)(13,31)(15,19)(16,34)(17,23)(18,35)(20,28)(21,30)(22,29)(27,33)(32,36);(5,9)(6,15)(7,8)(10,28)(11,30)(12,29)(13,23)(14,19)(16,27)(17,31)(18,36)(20,24)(21,25)(22,26)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^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,e^b=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301243);