Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,12,15,10)(8,11)(9,14)(13,17)(16,19,18,20);(1,2,3,4)(6,10,14,12)(11,19,13,20);(6,11)(7,15)(8,17)(10,19)(12,20)(13,14)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215601, #72234065, #72234081, … |
| maximal normal subgroup of | #72215601, #72234065, #72234081, … |
| maximal quotient of | #72213396, #72231140, #72231326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53852);