Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2093 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,19,12,20,33,18,13,21,10,17,23,36,9,16)(7,22,25,31,26,28,15,32,27,29,24,14,35,30,34,8);(2,4)(5,7,23,26,13,27,12,35)(6,14,36,8,21,31,20,32)(9,24,10,34,33,25,11,15)(16,28,17,29,18,30,19,22);(2,4)(5,8,23,31,13,32,12,14)(6,15,36,24,21,34,20,25)(7,19,26,16,27,17,35,18)(9,22,10,28,33,29,11,30)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=deh,d^2=hi,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250147, #72250148, #72250150, … |
| maximal normal subgroup of | #72250147, #72250148, #72250150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56608);