Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,16,10,15)(7,17,20,30,21,29,26,12)(8,22,23,13,25,32,24,33)(18,27,35,28,36,31,19,34);(1,2)(3,4)(5,7,10,20,11,21,9,26)(6,12,15,17,16,30,14,29)(8,18,24,35,25,36,23,19)(13,31,22,34,33,27,32,28);(2,4)(5,8)(6,13)(7,18,21,36)(9,24)(10,23)(11,25)(12,27,30,31)(14,22)(15,32)(16,33)(17,34,29,28)(19,26,35,20);(5,8)(6,13)(7,19,21,35)(9,23)(10,24)(11,25)(12,28,30,34)(14,32)(15,22)(16,33)(17,31,29,27)(18,26,36,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ef,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339204);