Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419794 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| 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 | |
985 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)(5,6,10,13,11,15,30,14)(7,16,19,33,21,32,20,12)(8,22,24,28,25,27,29,9)(17,35,36,34,31,18,23,26);(5,7,10,20,11,21,30,19)(6,12,13,32,15,33,14,16)(8,23,24,31,25,36,29,17)(9,26,22,18,28,34,27,35);(5,8,11,25)(6,9,15,28)(7,17,21,31)(10,29,30,24)(12,18,33,35)(13,27,14,22)(16,34,32,26)(19,23,20,36);(2,4)(5,9,11,28)(6,8,15,25)(7,18)(10,22,30,27)(12,31)(13,24,14,29)(16,23)(17,33)(19,26)(20,34)(21,35)(32,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336799);