Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1943 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(6,12,14,24)(7,16,19,18)(8,21)(9,23)(10,22)(13,17)(15,31,30,26)(20,27,33,34)(25,28)(29,36);(5,6,8,12,10,15,21,26,11,14,9,24,22,30,23,31)(7,17,16,32,20,29,34,25,19,36,18,35,33,13,27,28);(1,2,3,4)(5,7,9,16,10,20,23,34,11,19,8,18,22,33,21,27)(6,13,12,25,15,17,26,35,14,29,24,28,30,36,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239801, #72239807, #72239822, … |
| maximal normal subgroup of | #72239801, #72239807, #72239822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54301);