Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,15,10,16,11,14)(7,17,18,12,20,27,21,28)(8,13,23,31,24,32,25,30)(19,33,22,36,35,29,34,26);(1,2,3,4)(5,7,9,18,10,20,11,21)(6,12,14,17,16,28,15,27)(8,22,23,35,24,34,25,19)(13,29,30,36,32,33,31,26);(7,18)(9,11)(12,17)(14,15)(19,34)(20,21)(22,35)(23,25)(26,33)(27,28)(29,36)(30,31);(2,4)(5,8,10,24)(6,13)(7,19,20,35)(9,25,11,23)(12,26)(14,31)(15,30)(16,32)(17,33)(18,34,21,22)(27,29)(28,36)Polycyclic presentation:
2^9|a^2=fg,b^2=e,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,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, 343357);