Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3100 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,7,5,8,4,6)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15);(2,6)(3,4)(7,8)(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24);(2,7)(3,4)(6,8);(11,19)(15,23)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=g,b^2=egi,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239041, #72240781, #72279774, … |
| maximal normal subgroup of | #72239041, #72240781, #72279774, … |
| maximal quotient of | #72257795, #72285533, #72285656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105219);