Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,12,10,13,8,14)(7,15,16,19,17,11,18,20);(1,2,3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,19)(13,15)(14,20);(6,12)(8,9)(11,19)(13,14)(15,20)(16,18)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234152, #72234161, #72234170, … |
| maximal normal subgroup of | #72234152, #72234161, #72234170, … |
| maximal quotient of | #72234148, #72234564, #72261923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53818);