Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395409 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3468 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)(5,6,11,14,10,16,9,15)(7,17,19,12,20,28,24,27)(8,13,25,30,23,32,22,31)(18,33,21,26,34,29,35,36);(5,7)(6,12)(8,21)(9,24)(10,20)(11,19)(13,29)(14,28)(15,17)(16,27)(18,22)(23,35)(25,34)(26,31)(30,36)(32,33);(5,8,10,23)(6,13,16,32)(7,18,20,34)(9,25,11,22)(12,26,27,36)(14,31,15,30)(17,29,28,33)(19,35,24,21);(2,4)Polycyclic presentation:
2^9|a^2=ef,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302606);