Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13490 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2073 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,11)(7,17)(8,19)(9,12)(13,29)(14,16)(15,23)(18,31)(20,22)(21,35)(24,28)(25,26)(27,30)(32,33);(5,6,12,15,10,16,9,25)(7,18,27,34,22,31,21,36)(8,23,28,11,24,26,19,14)(13,30,29,17,32,35,33,20);(5,7,19,30,9,21,24,20,10,22,28,35,12,27,8,17)(6,13,26,34,15,29,14,31,16,32,23,36,25,33,11,18)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321333, #72321394, #72321421, … |
| maximal normal subgroup of | #72321333, #72321394, #72321421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56358);