Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(10,13)(11,15)(17,18,22,20)(19,24,23,21);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,24)(22,23);(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,21,23,24);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,24,23,21);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,21,22,24)(18,23,20,19)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #98640, #130306, #266792, … |
| maximal normal subgroup of | #98640, #130306, #266792, … |
| maximal quotient of | #531892, #531986, #532533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52453);