Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#386 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,5,10)(3,8,14,22,15,23,19,25)(4,16,17,26,18,9,20,24)(13,21,29,32,28,30,27,31);(1,3,6,19,7,15,5,14)(2,8,10,22,11,23,12,25)(4,13,20,29,18,28,17,27)(9,21,24,31,16,30,26,32);(1,4)(2,9)(3,13)(5,17)(6,20)(7,18)(8,21)(10,24)(11,16)(12,26)(14,27)(15,28)(19,29)(22,31)(23,30)(25,32);(1,5,7,6)(2,10,11,12)(3,14,15,19)(4,17,18,20)(8,22,23,25)(9,24,16,26)(13,27,28,29)(21,31,30,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146613, #146618, #148476, … |
| maximal normal subgroup of | #146613, #146618, #148476, … |
| maximal quotient of | #196310, #196311, #196316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14005);