Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185267 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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 | #62822 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,23,22,26,20,25)(19,24,29,30,27,31,28,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,23)(20,21)(24,30)(25,26)(28,29)(31,32);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,32)(26,31);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,27)(24,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,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 | #72268384, #72268393, #72268408, … |
| maximal normal subgroup of | #72268384, #72268393, #72268408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92552);