Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401009 |
| 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 | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1221 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)(7,18)(8,23)(9,27)(10,30)(11,17)(12,16)(13,20)(14,19)(15,28)(21,32)(22,36)(24,31)(25,34)(26,35)(29,33);(1,2,3,4)(5,7,10,20,11,22,12,21)(6,13,16,18,17,32,30,36)(8,24,25,9,19,28,26,29)(14,33,34,15,23,27,35,31);(5,8,11,19)(6,14,17,23)(7,9,22,29)(10,26,12,25)(13,31,32,15)(16,35,30,34)(18,27,36,33)(20,24,21,28);(5,9)(6,15,17,31)(7,19)(8,22)(10,28)(11,29)(12,24)(13,23,32,14)(16,27,30,33)(18,35,36,34)(20,26)(21,25)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=gh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321230);