Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22);(2,3)(9,11,14,12)(10,15,18,16)(13,20,21,19)(17,23,24,22);(1,2,4,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24);(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22);(1,3,4,2)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98332, #247407, #247408, … |
| maximal normal subgroup of | #98332, #247407, #247408, … |
| maximal quotient of | #521406, #522130, #522680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32331);