Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 65536 |
| 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 | #2699 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,15)(11,13)(18,22)(19,25)(20,28)(30,32);(2,4)(7,8)(9,10,14,15)(11,12,13,16);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,16,15,12)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,4,6,2)(3,7,5,8)(9,12,14,16)(10,11,15,13)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,5,6,3)(2,7,4,8)(9,13,14,11)(10,16,15,12)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|b^2=h,c^2=fgh,d^2=fgh,e^2=f,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 | #132579, #487340, #487932, … |
| maximal normal subgroup of | #132579, #487340, #487932, … |
| maximal quotient of | #582192, #582434, #582641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45941);