Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24);(5,6,8,7)(11,17)(13,21)(15,19)(16,20);(1,2,3,4)(5,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| Frattini subgroup of | #72266409 |
| maximal subgroup of | #97457, #132048, #133758, … |
| maximal normal subgroup of | #97457, #132048, #133758, … |
| maximal quotient of | #515329, #515330, #515558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28622);