Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2085 |
| 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 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,16,11,14,10,27)(7,17,20,31,18,12,26,30)(8,21,24,32,25,13,23,15)(19,34,36,33,22,28,35,29);(1,2)(3,4)(5,7,10,20,11,18,9,26)(6,12,27,30,14,17,16,31)(8,22,23,35,25,19,24,36)(13,28,32,29,21,34,15,33);(2,4)(6,13,14,21)(7,18)(8,23)(9,10)(12,28,17,34)(15,16,32,27)(19,35)(22,36)(24,25)(29,30,33,31)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=d,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239863, #72239866, #72240850, … |
| maximal normal subgroup of | #72239863, #72239866, #72240850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55944);