Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1080 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)(6,10,12,19)(7,14)(8,17)(11,20,13,15)(16,18);(5,6)(7,15)(8,11)(9,12)(10,18)(13,17)(14,19)(16,20);(2,4)(7,16)(10,19)(14,18)(15,20);(6,11,12,13)(7,16)(10,20,19,15)(14,18)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224588, C4×#33688, #72249864, … |
| maximal normal subgroup of | #72224588, C4×#33688, #72249864, … |
| maximal quotient of | #72224575, #72242475, #72258425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402927);