Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
798 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)(5,6,9,14,10,12,26,15)(7,16,19,29,20,27,17,11)(8,21,24,32,25,13,23,31)(18,33,35,28,34,30,22,36);(5,7,10,20)(6,11,12,29)(8,22,25,35)(9,17,26,19)(13,30,21,33)(14,27,15,16)(18,24,34,23)(28,32,36,31);(6,12)(7,17)(8,23)(9,26)(11,27)(13,31)(16,29)(18,34)(19,20)(21,32)(24,25)(28,36);(2,4)(5,8)(6,13,12,21)(7,18)(9,24)(10,25)(11,28,29,36)(14,31,15,32)(16,30,27,33)(17,22)(19,35)(20,34)(23,26)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386427);