Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,11)(7,15)(8,9)(13,14)(16,17)(18,19);(1,3,4,2)(5,6)(7,12)(8,14)(9,11)(10,13)(15,18)(16,20)(17,19);(6,11,13,14)(12,18,20,19);(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72311981, C4×#24472, #72312202 |
| maximal normal subgroup of | #72311981, C4×#24472, #72312202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272643);