Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135928 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,17,28,19,9,15,13,16,29,33,12,7,10,20)(8,21,25,22,36,18,24,31,27,14,30,32,26,34,23,35);(1,2)(3,4)(5,7,12,6,13,19,28,16)(8,22,26,31,27,32,36,35)(9,20,11,17,10,15,29,33)(14,23,34,30,21,24,18,25);(2,4)(5,8,9,23,12,26,11,30,13,27,10,24,28,36,29,25)(6,14,15,31,7,18,17,22,16,21,20,35,19,34,33,32)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=h,f^2=h,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=fhi,g^b=ghi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250001, #72250003, #72250418, … |
| maximal normal subgroup of | #72250001, #72250003, #72250418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56276);