Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1320 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)(5,6,9,13,10,15,25,14)(7,16,19,28,20,27,17,11)(8,12,22,30,23,31,24,32)(18,33,34,26,35,29,21,36);(5,7)(6,11)(8,21)(9,17)(10,20)(12,29)(13,27)(14,16)(15,28)(18,24)(19,25)(22,35)(23,34)(26,30)(31,33)(32,36);(7,17)(8,22)(9,25)(11,16)(12,30)(13,14)(18,35)(19,20)(23,24)(27,28)(29,33)(31,32);(2,4)(5,8)(6,12)(7,18)(9,24)(10,23)(11,26)(13,32)(14,30)(15,31)(16,33)(17,34)(19,21)(20,35)(22,25)(27,29)(28,36)Polycyclic presentation:
2^9|a^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383340);