Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,14,16)(11,17,18,12)(13,21,22,15)(19,20,24,23);(5,6,8,7)(11,18)(12,17)(19,24)(20,23);(1,2)(3,4)(5,7,8,6)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,23)(22,24);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23);(9,13,14,22)(10,15,16,21)(11,19,18,24)(12,20,17,23)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97692, #133901, #134101, … |
| maximal normal subgroup of | #97692, #133901, #134101, … |
| maximal quotient of | #516392, #516394, #516579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28296);