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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
995 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);(6,10)(8,11)(13,14,17,20)(15,18,23,21)(16,24,25,19)(22,27,28,26);(1,2,3,4)(6,10)(7,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28);(5,7,9,12)(6,11,10,8)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(5,8,9,11)(6,12,10,7)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)Polycyclic presentation:
2^8|a^2=f,b^2=h,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 | #132204, #248449, #248450, … |
| maximal normal subgroup of | #132204, #248449, #248450, … |
| maximal quotient of | #544529, #545086, #545504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35801);