Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235462 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1474 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,15)(14,16)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(2,7)(4,5)(8,11)(9,10)(13,14)(15,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,3,6,12)(2,8,10,16)(4,13,5,14)(7,15,9,11)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31);(3,12)(8,16)(11,15)(13,14)Polycyclic presentation:
2^9|c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72249774, #72249987, #72305970, … |
| maximal normal subgroup of | #72249774, #72249987, #72305970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142662);