Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2711 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)(7,10)(8,15,17,11)(9,12)(13,18,16,20)(14,19);(2,4)(5,7)(6,10)(8,16,17,13)(9,14)(11,20,15,18)(12,19);(5,8)(6,11)(7,13)(9,17)(10,18)(12,15)(14,16)(19,20);(2,4)(5,6)(7,10)(8,11)(9,12)(13,18)(14,19)(15,17)(16,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72233009, #72233072, #72233100, … |
| maximal normal subgroup of | #72233009, #72233072, #72233100, … |
| maximal quotient of | #72232995, #72233066, #72234114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271741);