Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393900 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,12,16,13,17,22,15)(11,18,20,14,21,23,19,24);(1,2,3,4)(9,11,12,20,13,21,22,19)(10,14,15,18,17,24,16,23);(10,15)(11,19)(12,22)(16,17)(18,23)(20,21);(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239135, #72239142, #72239146 |
| maximal normal subgroup of | #72239135, #72239142, #72239146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302768);