Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187934 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #62820 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,19)(4,14)(8,18,20,31)(9,17,22,29)(12,23)(15,27)(21,28,32,30)(25,26);(1,2)(3,8)(4,7)(5,10)(6,9)(11,20)(12,18)(13,21)(14,19)(15,17)(16,22)(23,31)(24,32)(25,30)(26,28)(27,29);(3,11)(4,15)(7,17)(8,20)(12,25)(13,24)(14,27)(18,30)(19,29)(21,32)(23,26)(28,31);(1,3,6,13)(2,8,9,21)(4,12,15,26)(5,11,16,24)(7,18,17,28)(10,20,22,32)(14,23,27,25)(19,31,29,30)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,e^a=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72266859, #72266861, #72266864, … |
| maximal normal subgroup of | #72266859, #72266861, #72266864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95133);