Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 |
| prime radical | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3324 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)(9,10,14,17)(11,18,16,13)(12,20,21,19)(15,23,24,22);(5,6)(7,8)(9,11,13,10,14,16,18,17)(12,19,22,15,21,20,23,24);(2,4)(6,8)(9,12,13,22,14,21,18,23)(10,15,16,20,17,24,11,19);(6,8);(12,21)(15,24)(19,20)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308227, #72308238, #72308373, … |
| maximal normal subgroup of | #72308227, #72308238, #72308373, … |
| maximal quotient of | #72315686 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346152);