Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
828 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)(5,6,9,15,11,16,10,14)(7,17,20,12,21,28,19,29)(8,22,26,31,24,13,25,32)(18,33,23,27,34,30,35,36);(2,4)(5,7,10,20,11,21,9,19)(6,12,15,17,16,29,14,28)(8,23,26,18,24,35,25,34)(13,30,31,36,22,33,32,27);(5,8)(6,13)(7,18)(9,25)(10,26)(11,24)(12,27)(14,32)(15,31)(16,22)(17,30)(19,23)(20,35)(21,34)(28,33)(29,36);(6,14,16,15)(8,24)(12,28,29,17)(13,31,22,32)(18,34)(23,35)(25,26)(27,30,36,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=di,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, 316812);