Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,10,14,26,13)(7,16,18,11,19,31,20,28)(8,21,24,29,25,12,23,30)(17,33,35,27,22,32,34,36);(1,2)(3,4)(5,7)(6,11,14,28)(8,22)(9,18)(10,19)(12,27,21,36)(13,16,15,31)(17,25)(20,26)(23,35)(24,34)(29,33,30,32);(2,4)(5,8)(6,12)(7,17)(9,23)(10,25)(11,27)(13,30)(14,21)(15,29)(16,32)(18,34)(19,22)(20,35)(24,26)(28,36)(31,33);(6,13)(8,23)(9,26)(11,16)(12,21)(14,15)(17,34)(18,20)(22,35)(24,25)(27,36)(28,31)Polycyclic presentation:
2^9|a^2=ef,b^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 363697);