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) = C28 |
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 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,7)(10,11)(13,14,17,20)(15,16,19,18);(5,6,9,7)(8,11,12,10)(15,19)(16,18);(1,2)(3,4)(13,15,17,19)(14,18,20,16);(5,7,9,6)(8,11,12,10);(5,8,9,12)(6,10,7,11)(13,16,17,18)(14,19,20,15)Polycyclic presentation:
2^8|a^2=h,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 | #98316, #247411, #247412, … |
| maximal normal subgroup of | #98316, #247411, #247412, … |
| maximal quotient of | #521406, #522181, #522582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32458);