Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,8,11,14,9,13)(7,15,17,12,18,19,16,20);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,19)(14,20);(2,4)(6,8)(9,10)(12,19)(13,14)(15,20)(16,17);(5,8,9,6,11,13,10,14)(7,15,17,12,18,19,16,20)Polycyclic presentation:
2^9|a^2=h,d^2=efh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224752, #72224754, #72224755, … |
| maximal normal subgroup of | #72224752, #72224754, #72224755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289086);