Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #399 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10,13,20,15,19,14,18)(11,21,12,24,23,17,22,16);(9,11,14,22,15,23,13,12)(10,16,18,17,19,24,20,21);(1,2,3,4)(9,12,13,23,15,22,14,11)(10,17,18,24,19,21,20,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72215689, #72216103, #72238740, … |
| maximal normal subgroup of | #72215689, #72216103, #72238740, … |
| maximal quotient of | #72215669, #72239020, #72239100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31099);