Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,15)(7,16,19,30)(8,20)(9,11,25,14)(12,24,27,17)(13,23)(18,31)(21,29)(22,26)(28,32)(33,35)(34,36);(2,4)(6,11,15,14)(7,17,19,24)(12,27)(13,26,20,29)(16,30)(18,32,34,33)(28,35)(31,36);(5,7)(6,12)(8,18)(9,24)(10,19)(11,16)(13,28)(14,30)(15,27)(17,25)(20,35)(21,33)(22,32)(23,34)(26,31)(29,36);(5,8)(6,13)(7,18)(9,21)(10,23)(11,26)(12,28)(14,29)(15,20)(16,31)(17,32)(19,34)(22,25)(24,33)(27,35)(30,36)Polycyclic presentation:
2^9|a^2=ef,b^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250134, #72257147, #72257214 |
| maximal normal subgroup of | #72250134, #72257147, #72257214 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312965);