Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23895 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1957 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)(7,13)(8,11)(9,12)(10,15)(14,17)(16,18)(19,20);(1,3,4,2)(10,13)(17,19);(5,7)(6,10,12,13)(8,14)(9,15)(11,17,18,19)(16,20);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72242791, C4×#23895, #72242909, … |
| maximal normal subgroup of | #72242791, C4×#23895, #72245896, … |
| maximal quotient of | #72242909, #72247306, #72282647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275268);