Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(10,15)(12,13)(16,23)(17,18)(19,21)(22,24);(9,10,13,18,14,17,12,15)(11,16,19,23,20,24,21,22);(5,6,7,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72219669, #72239567, #72239574, … |
| maximal normal subgroup of | #72219669, #72239567, #72239574, … |
| maximal quotient of | #72219659, #72239222, #72239224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30678);