Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105783 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13,16,8)(4,7)(5,12)(6,10)(9,18)(11,19)(14,24,28,25)(15,29,31,20)(17,21)(22,32)(23,27,26,30);(2,7,11,22)(3,14)(4,17)(8,23,24,29)(9,10,21,12)(13,26,25,20)(15,30)(16,28)(18,32)(27,31);(1,3)(2,8)(4,15)(5,14)(6,16)(7,20)(9,23)(10,13)(11,24)(12,25)(17,31)(18,27)(19,28)(21,29)(22,26)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,c^b=ch,d^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233291, #72233308, #72233311, … |
| maximal normal subgroup of | #72233291, #72233308, #72233311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12979);