Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,12,26,10,16,11,17)(7,13,25,34,20,29,21,30)(8,15,24,32,9,14,23,33)(18,35,19,31,36,27,22,28);(1,2,4,3)(5,7,10,20)(6,13,16,29)(8,22,9,19)(11,25,12,21)(14,31,15,28)(17,34,26,30)(18,24,36,23)(27,33,35,32);(2,3)(5,8)(6,14)(7,18)(9,10)(11,24)(12,23)(13,27)(15,16)(17,33)(19,21)(20,36)(22,25)(26,32)(28,30)(29,35)(31,34);(1,3,4,2)(5,9,10,8)(6,15)(7,19,20,22)(11,24,12,23)(13,28)(14,16)(17,33)(18,25,36,21)(26,32)(27,30)(29,31)(34,35)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368497);