Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,16)(13,24)(14,19)(17,21)(18,23)(20,22);(2,4)(10,15,19,18)(11,12)(13,22)(16,20,17,24)(21,23);(10,16)(11,21)(12,23)(13,22)(15,20)(17,19)(18,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216453, #72216458, #72216468, … |
| maximal normal subgroup of | #72216453, #72216458, #72216468, … |
| maximal quotient of | #72264879, #72264903, #72265129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53995);