Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,22,25,30)(9,15)(10,14)(11,16)(12,18)(13,24,29,23)(19,33,35,26)(20,28)(21,27)(31,34,36,32);(5,7,9,18,11,21,10,20)(6,12,14,17,16,28,15,27)(8,19,24,32,25,35,23,34)(13,26,30,31,29,33,22,36);(1,2,3,4)(7,18)(8,23)(9,10)(12,17)(13,22)(14,15)(20,21)(24,25)(27,28)(29,30)(31,36)(32,34);(5,8)(6,13)(7,19)(9,24)(10,23)(11,25)(12,26)(14,30)(15,22)(16,29)(17,31)(18,32)(20,34)(21,35)(27,36)(28,33)Polycyclic presentation:
2^9|a^2=eg,b^2=gi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374435);