Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1011 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)(5,6,9,16,11,15,10,14)(7,17,19,29,20,12,25,28)(8,21,24,32,23,31,26,13)(18,30,22,27,34,36,35,33);(5,7,10,19,11,20,9,25)(6,12,14,28,15,17,16,29)(8,22,24,18,23,35,26,34)(13,30,21,33,32,36,31,27);(2,4)(5,8)(6,13)(7,18)(9,26)(10,24)(11,23)(12,27)(14,21)(15,32)(16,31)(17,33)(19,35)(20,34)(22,25)(28,30)(29,36);(6,14,15,16)(8,23)(12,28,17,29)(13,31,32,21)(18,34)(22,35)(24,26)(27,36,33,30)Polycyclic presentation:
2^9|a^2=f,b^2=eh,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280664 |
| maximal normal subgroup of | #72280664 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313314);