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 | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12,10,16,11,15)(7,17,20,30,18,27,21,13)(8,22,24,14,25,28,26,32)(19,29,23,33,34,36,35,31);(1,2,3,4)(6,12,16,15)(7,18)(13,27,30,17)(14,28,32,22)(19,34)(20,21)(23,35)(29,31,36,33);(2,4)(5,7,10,18)(6,13,16,30)(8,23,25,35)(9,21,11,20)(12,27,15,17)(14,31,32,33)(19,26,34,24)(22,29,28,36);(5,8,9,24,10,25,11,26)(6,14,12,28,16,32,15,22)(7,19,20,23,18,34,21,35)(13,29,17,33,30,36,27,31)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=i,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, 373045);