Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1762 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)(9,10)(11,19,22,15)(12,17)(13,18)(14,16)(20,24,21,23);(5,6)(7,8)(9,11,13,20,14,22,12,21)(10,15,18,23,16,19,17,24);(1,2,3,4)(6,8)(11,20)(12,13)(15,23)(17,18)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72260864, #72260897, #72260912 |
| maximal normal subgroup of | #72260864, #72260897, #72260912 |
| maximal quotient of | #72261112 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31152);