Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13545 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2101 |
| 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 | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,12,16,18,21,29,28,10,17,11,20,13,7,9,15)(8,14,26,31,34,36,23,30,24,32,25,33,27,19,35,22);(1,2,4,3)(5,7,18,17,10,21,13,6)(8,22,34,33,24,30,27,31)(9,28,12,16,29,15,11,20)(14,25,19,35,32,26,36,23);(1,3,4,2)(5,8,9,23,13,27,11,26,10,24,29,35,18,34,12,25)(6,14,15,30,7,19,20,31,17,32,28,22,21,36,16,33)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=h,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239853, #72239863, #72240849, … |
| maximal normal subgroup of | #72239853, #72239863, #72240849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56274);