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 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,11,18,28,17)(7,19,22,35,23,32,36,31)(8,15,24,14,25,12,9,26)(13,29,33,27,34,21,30,20);(1,2)(3,4)(6,12,18,15)(7,20,23,27)(8,24)(9,25)(10,28)(13,30)(14,17,26,16)(21,36,29,22)(31,35)(33,34);(2,4)(5,7,8,21,10,22,24,20,11,23,25,29,28,36,9,27)(6,13,14,31,17,30,15,32,18,34,26,35,16,33,12,19)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239850, #72240848, #72240849, … |
| maximal normal subgroup of | #72239850, #72240848, #72240849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56316);