Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,12,9,11,17,19)(7,13,15,20,14,10,16,18);(2,4)(5,7)(6,10)(8,16)(9,14)(11,13)(12,20)(15,17)(18,19);(10,13)(18,20);(2,4)(6,11)(8,17)(10,13)(15,16)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216073, #72219299, #72219306, … |
| maximal normal subgroup of | #72216073, #72219299, #72219306, … |
| maximal quotient of | #72232717, #72233223, #72235062, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266789);