Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187940 |
| 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 | |
2494 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)(3,11)(4,14)(8,20,21,18)(9,17,23,30)(13,25)(15,28)(22,31,32,29);(1,2)(3,8)(4,7)(5,10)(6,9)(11,21)(12,18)(13,22)(14,19)(15,17)(16,23)(20,24)(25,32)(26,31)(27,29)(28,30);(3,11)(4,15)(7,17)(8,21)(12,26)(13,25)(14,28)(18,31)(19,30)(20,29)(22,32)(24,27);(1,3,6,13)(2,8,9,22)(4,12,15,27)(5,11,16,25)(7,18,17,29)(10,21,23,32)(14,24,28,26)(19,20,30,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234508, #72234516, #72234937, … |
| maximal normal subgroup of | #72234508, #72234516, #72234937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95139);