Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,8,25,17)(7,18,22,33,24,34,21,35)(9,16,30,14,31,28,11,27)(13,23,32,29,26,36,19,20);(2,4)(5,7,11,23,10,22,9,29,12,24,30,36,25,21,31,20)(6,13,16,33,17,19,27,18,8,26,28,35,15,32,14,34);(5,8)(6,12)(7,19,24,32)(9,27)(10,15)(11,16)(13,22,26,21)(14,31)(17,25)(18,23,34,36)(20,35,29,33)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239801, #72239814, #72239844, … |
| maximal normal subgroup of | #72239801, #72239814, #72239844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54636);