Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
3222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,16,12,21)(11,18,19,15,20,24,22,23);(5,6,7,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,23)(17,24)(18,21);(2,4)(10,16)(12,13)(18,24)(19,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237175, #72237176, #72237181, … |
| maximal normal subgroup of | #72237175, #72237176, #72237181, … |
| maximal quotient of | #72239039, #72239437, #72240288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30783);