Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #381 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(10,14,16,20)(15,22,17,23);(5,6)(7,8)(9,10)(11,17,19,15)(12,20)(13,16)(14,21)(18,22,24,23);(1,2,3,4)(6,8)(9,11)(10,15)(12,18)(13,19)(14,22)(16,17)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fi,g^b=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235513 |
| maximal normal subgroup of | #72235513 |
| maximal quotient of | #72235541 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31225);