Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #396 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#396 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1422 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)(5,6,11,15,10,16,9,14)(7,17,20,12,21,29,19,28)(8,22,24,32,25,13,26,31)(18,33,23,36,35,30,34,27);(5,7,11,20,10,21,9,19)(6,12,15,29,16,28,14,17)(8,23,24,35,25,34,26,18)(13,30,31,27,22,33,32,36);(5,8,9,24,10,25,11,26)(6,13,15,32,16,22,14,31)(7,18,20,34,21,35,19,23)(12,27,17,33,28,36,29,30);(5,9,10,11)(6,14,16,15)(7,19,21,20)(8,24,25,26)(12,17,28,29)(13,31,22,32)(18,23,35,34)(27,33,36,30)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265523);