Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C42×D4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3,4,6,8,7)(9,10)(11,18)(12,20)(13,16)(14,17)(15,22)(19,21)(23,24);(2,6)(3,7)(9,11,13,10,14,16,18,17)(12,19,21,15,22,23,24,20);(9,12,13,21,14,22,18,24)(10,15,16,23,17,20,11,19);(1,3,5,6,4,7,8,2)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ch,d^a=di,e^a=eh,d^b=di,e^b=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72309486, #72309492, #72309508, … |
| maximal normal subgroup of | #72309486, #72309492, #72309508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91224);