Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,8)(13,14,18,16)(15,21,23,19)(17,20,25,24)(22,27,28,26);(10,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27);(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| Fitting subgroup of | #14336⋊C3 |
| maximal subgroup of | #108240, #111893, #111897, … |
| maximal normal subgroup of | #108240, #111893, #111897, … |
| maximal quotient of | #153932, #153945, #154120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7324);