Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185475 |
| 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 | C22×#555 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15);(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72293112, #72293173, #72293178, … |
| maximal normal subgroup of | #72293112, #72293173, #72293178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92673);