Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,15,11,17,10,16)(7,8,20,23,21,24,19,13)(9,25,29,33,30,14,28,32)(18,34,35,36,26,22,31,27);(5,7,10,19,11,21,12,20)(6,13,15,8,17,23,16,24)(9,26,28,35,30,18,29,31)(14,27,32,34,25,36,33,22);(5,8,10,23,11,24,12,13)(6,7,15,19,17,21,16,20)(9,27,28,34,30,36,29,22)(14,31,32,26,25,35,33,18);(2,4)(5,9,10,29,11,30,12,28)(6,14,16,32,17,25,15,33)(7,18,20,31,21,26,19,35)(8,22,23,36,24,34,13,27)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=e,d^2=f,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,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386478);