Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,7)(10,15)(11,13)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(1,2,5,4)(3,8,7,6)(9,10,14,15)(11,12,13,16);(3,7)(6,8)(9,11,14,13)(10,16,15,12)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,3,5,7)(2,6,4,8)(9,12,14,16)(10,11,15,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,16,15,12)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=g,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132568, #132764, #487322, … |
| maximal normal subgroup of | #132568, #132764, #487322, … |
| maximal quotient of | #582238, #582457, #582620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46019);