Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185273 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62820 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,21)(3,12)(4,15)(8,22,25,29)(9,19,11,16)(13,26)(17,31)(20,23,32,24);(1,2)(3,8)(4,16)(5,9)(6,11)(7,17)(10,18)(12,24)(13,23)(14,29)(15,21)(19,31)(20,28)(22,30)(25,26)(27,32);(3,13)(4,15)(7,19)(8,23)(12,26)(14,30)(16,21)(17,31)(20,32)(22,29)(24,25)(27,28);(1,3)(2,8)(4,14)(5,12)(6,13)(7,20)(9,24)(10,25)(11,23)(15,27)(16,29)(17,28)(18,26)(19,22)(21,32)(30,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234505, #72234937, #72234940, … |
| maximal normal subgroup of | #72234505, #72234937, #72234940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92469);