Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,16,12,25)(7,17,20,34,19,33,26,27)(8,14,9,23,21,31,22,32)(13,24,29,35,28,36,30,18);(1,2,3,4)(5,7,11,19)(6,13,16,28)(8,18,21,35)(9,24,22,36)(10,20,12,26)(14,27,31,34)(15,29,25,30)(17,32,33,23);(5,8,10,9,11,21,12,22)(6,14,15,23,16,31,25,32)(7,18,20,24,19,35,26,36)(13,27,29,17,28,34,30,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dh,e^b=ei,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54412);