Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,16,18,29)(8,20)(9,25)(10,27)(11,15)(12,17,28,21)(13,24)(14,23)(19,35,34,32)(22,26)(30,36,31,33);(5,7,9,17,10,19,26,33,11,18,8,21,23,34,24,36)(6,12,13,29,14,30,20,35,15,28,22,16,27,31,25,32);(1,2,3,4)(5,6)(7,12)(8,22)(9,13)(10,14)(11,15)(16,21)(17,29)(18,28)(19,30)(20,26)(23,27)(24,25)(31,34)(32,36)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=di,e^b=ei,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224893, #72224931, #72224952, … |
| maximal normal subgroup of | #72224893, #72224931, #72224952, … |
| maximal quotient of | #72224886 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54604);