Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,30)(23,25,29,26)(24,28,33,35)(31,34,36,32);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,22,27,29,25,30)(24,31,32,28,33,36,34,35);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,28)(23,31)(25,34)(26,32)(27,33)(29,36)(30,35);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,29,30,23)(24,32,33,34)(28,31,35,36);(8,15)(11,18)(14,19)(17,20)(24,33)(28,35)(31,36)(32,34)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283144, #72283615, #72284663, … |
| maximal normal subgroup of | #72283144, #72283615, #72284663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476789);