Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459488 |
| 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 | |
1069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,10,16,11,13)(7,17,20,12,18,27,21,28)(8,22,23,14,24,30,25,29)(19,33,35,31,32,26,34,36);(1,2,3,4)(5,7)(6,12,16,28)(8,19)(9,20)(10,18)(11,21)(13,17,15,27)(14,26,29,33)(22,31,30,36)(23,35)(24,32)(25,34);(2,4)(6,13)(7,18)(8,23)(9,11)(12,17)(14,29)(15,16)(19,34)(24,25)(26,33)(27,28)(32,35);(5,8,9,23,10,24,11,25)(6,14,15,30,16,29,13,22)(7,19,20,35,18,32,21,34)(12,26,27,36,28,33,17,31)Polycyclic presentation:
2^9|a^2=f,b^2=gi,d^2=f,e^2=i,f^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,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373044);