Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,19,14,18,15,16)(11,20,22,23,12,17,21,24);(5,6)(7,8)(10,16)(11,21)(12,22)(13,15)(17,20)(18,19);(1,2,3,4)(6,8)(9,11,15,21,14,12,13,22)(10,17,16,23,18,20,19,24);(6,8)(9,12,13,21,14,11,15,22)(10,17,19,24,18,20,16,23);(10,18)(16,19)(17,20)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,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=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284383, #72312885, #72312994, … |
| maximal normal subgroup of | #72284383, #72312885, #72312994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351555);