Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,14)(7,12,26,17,20,29,19,28)(8,21,25,22,23,30,24,13)(18,31,35,27,33,32,34,36);(1,2,3,4)(5,7,11,20)(6,12,16,29)(8,18,23,33)(9,19,10,26)(13,27,22,36)(14,28,15,17)(21,32,30,31)(24,34,25,35);(2,4)(5,6,11,16)(7,17,20,28)(8,22,23,13)(9,15,10,14)(12,19,29,26)(18,32,33,31)(21,24,30,25)(27,34,36,35);(5,8,9,25,11,23,10,24)(6,13,14,21,16,22,15,30)(7,18,19,35,20,33,26,34)(12,27,28,32,29,36,17,31)Polycyclic presentation:
2^9|a^2=fi,b^2=egh,c^2=i,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380333);