Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,13)(7,14,12,9)(8,11,16,18)(15,20,19,17)(21,22)(23,29)(24,26)(25,33)(27,32)(28,30)(31,35)(34,36);(5,7,9,6,10,12,14,13)(8,15,17,11,16,19,20,18)(22,26)(27,34)(28,33)(29,35);(5,8,10,16)(6,11,13,18)(7,15,12,19)(9,17,14,20)(21,23,30,36,24,31,25,32)(22,27,33,35,26,34,28,29);(1,3,4,2)(8,16)(11,18)(15,19)(17,20)Polycyclic presentation:
2^9|a^2=g,b^2=ei,c^2=fhi,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120295);