Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,15)(11,17)(13,23)(18,19)(20,24)(21,22);(9,10)(11,16)(12,20)(13,18)(14,19)(15,23)(17,21)(22,24);(9,11,13,17,14,20,23,24)(10,16,18,21,19,12,15,22);(9,12)(10,17)(11,15)(13,22)(14,16)(18,20)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=g,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,e^d=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72216388, #72216391, #72216392, … |
| maximal normal subgroup of | #72216388, #72216391, #72216392, … |
| maximal quotient of | #72216394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132690);