Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,12,15,11,9,10,16)(7,14,18,20,8,19,17,13);(5,7,10,17,11,8,12,18)(6,13,15,14,9,20,16,19);(2,4)(5,8)(6,14)(7,11)(9,19)(10,18)(12,17)(13,16)(15,20);(2,4)(5,9)(6,11)(7,14)(8,19)(10,15)(12,16)(13,17)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72227424, #72227999, #72228011, … |
| maximal normal subgroup of | #72227424, #72227999, #72228011, … |
| maximal quotient of | #72268631 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290379);