Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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×#555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,10)(7,8,12,11)(13,14,18,22)(15,17,21,19)(16,20,26,28)(23,25,24,27);(6,10)(8,11)(15,21)(17,19)(23,24)(25,27);(1,2,3,4)(6,10)(7,12)(13,15,18,21)(14,19,22,17)(16,24,26,23)(20,25,28,27);(5,7,9,12)(6,11,10,8)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,24)(22,28);(5,8,9,11)(6,12,10,7)(13,17,18,19)(14,21,22,15)(16,25,26,27)(20,24,28,23)Polycyclic presentation:
2^8|a^2=fh,c^2=fg,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132210, #248374, #248375, … |
| maximal normal subgroup of | #132210, #248374, #248375, … |
| maximal quotient of | #544526, #545086, #545501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35669);