Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15)(7,16,20,27)(8,13)(9,12,24,14)(11,17,25,19)(18,31)(21,28)(22,29)(23,30)(26,33)(32,35)(34,36);(1,2)(3,4)(5,7)(6,11)(8,18)(9,17)(10,20)(12,16)(13,26)(14,27)(15,25)(19,24)(21,32)(22,33)(23,34)(28,31)(29,36)(30,35);(2,4)(6,12)(7,17)(8,21)(9,24)(11,25)(14,15)(19,20)(22,23)(26,36)(28,29)(31,35)(32,33);(5,8,9,21,10,23,24,22)(6,13,14,29,15,30,12,28)(7,18,19,33,20,34,17,32)(11,26,16,31,25,35,27,36)Polycyclic presentation:
2^9|a^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383855);