Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1739 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)(5,6)(7,14)(8,17)(9,12)(10,13)(11,15)(16,19)(18,20);(1,3,4,2)(7,15)(8,9)(11,19)(12,17)(14,20)(16,18);(5,7,9,15,10,16,8,18)(6,11,12,19,13,20,17,14);(5,6)(7,11)(8,17)(9,12)(10,13)(14,18)(15,19)(16,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72221416, C4×#23872, #72261241, … |
| maximal normal subgroup of | C4×#23872, #72261241, #72261250, … |
| maximal quotient of | #72221416, #72283187 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275298);