Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2794 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)(5,6,8,7)(9,10,14,18,13,16,22,23)(11,15,19,12,17,21,24,20);(6,7)(9,11,14,19,13,17,22,24)(10,15,18,12,16,21,23,20);(2,4)(10,16)(11,17)(18,23)(19,24);(5,7,8,6)(9,12)(10,17)(11,16)(13,20)(14,21)(15,22)(18,24)(19,23);(10,16)(12,20)(15,21)(18,23)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=df,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72312866, #72312881, #72313006 |
| maximal normal subgroup of | #72312866, #72312881, #72313006 |
| maximal quotient of | #72315823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289019);