Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,19,23,21,24,26,30,20,22,25,28,27,29,31,32);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(18,22)(19,25)(24,29)(26,31);(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fgi,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#126586, #72211373, #72211432, … |
| maximal normal subgroup of | C2×#126586, #72211373, #72211432, … |
| maximal quotient of | C2×#126586 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33782);