Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10,13,18,14,19,23,17)(11,20,12,16,22,15,24,21);(2,4)(9,10)(11,21)(12,15)(13,17)(14,19)(16,22)(18,23)(20,24);(5,6,7,8)(9,11,13,12,14,22,23,24)(10,15,18,21,19,20,17,16);(9,12,23,11,14,24,13,22)(10,16,17,20,19,21,18,15);(9,13,14,23)(10,17,19,18)(11,12,22,24)(15,16,20,21)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224791, #72224952, #72226670, … |
| maximal normal subgroup of | #72224791, #72224952, #72226670, … |
| maximal quotient of | #72224776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372831);