Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| 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 | |
1048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,13)(7,17,18,12,21,28,20,26)(8,14,24,29,25,30,23,31)(19,32,34,36,22,35,33,27);(1,2)(3,4)(5,7,10,18,11,21,9,20)(6,12,15,28,16,26,13,17)(8,22,24,33,25,19,23,34)(14,27,29,32,30,36,31,35);(2,4)(6,13)(7,18)(8,23)(9,10)(12,26)(14,30)(15,16)(20,21)(24,25)(27,32)(33,34)(35,36);(5,8,11,25)(6,14,16,30)(7,19,21,22)(9,24,10,23)(12,27,26,36)(13,29,15,31)(17,32,28,35)(18,33,20,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384198);