Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C22×C4×D4 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11,7,3,12,10)(4,13,16,22,17,8,20,23)(5,9,19,25,18,14,26,24)(15,27,29,31,28,21,30,32);(1,3)(2,7)(4,13)(5,14)(6,10)(8,17)(9,18)(11,12)(15,27)(16,22)(19,24)(20,23)(21,28)(25,26)(29,31)(30,32);(1,4,6,16,7,17,12,20)(2,8,11,23,3,13,10,22)(5,15,19,29,18,28,26,30)(9,21,25,32,14,27,24,31);(1,5)(2,9)(3,14)(4,15)(6,19)(7,18)(8,21)(10,24)(11,25)(12,26)(13,27)(16,29)(17,28)(20,30)(22,31)(23,32);(5,18)(9,14)(15,28)(19,26)(21,27)(24,25)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=g,c^2=g,f^2=h,g^2=h,b^a=bf,c^a=ch,f^a=fh,f^b=fh,e^d=eh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60184⋊C3 |
| maximal subgroup of | #495737, #495738, #6383886, … |
| maximal normal subgroup of | #495737, #495738, #6383886, … |
| maximal quotient of | #6344531, #6344532, #6344533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53172);