Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,32,11,15,34,16)(7,17,20,24,21,12,19,28)(8,22,33,35,25,18,26,13)(9,27,30,23,31,14,29,36);(5,7,10,19,11,21,34,20)(6,12,32,24,15,17,16,28)(8,23,33,27,25,36,26,14)(9,13,29,22,31,35,30,18);(5,8,10,33,11,25,34,26)(6,13,32,22,15,35,16,18)(7,14,19,23,21,27,20,36)(9,28,29,12,31,24,30,17);(2,4)(5,9,11,31)(6,14,15,27)(7,18,21,22)(8,24,25,28)(10,30,34,29)(12,26,17,33)(13,19,35,20)(16,23,32,36)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386113);