Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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×#547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1155 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)(13,14)(15,22)(16,19)(17,24)(18,20)(21,26)(23,27)(25,28);(6,9)(7,11)(13,15,19,18)(14,20,16,22)(17,23,26,25)(21,27,24,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,16,19,14)(15,20,18,22)(17,24,26,21)(23,27,25,28);(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(5,7)(6,10)(8,11)(9,12)(13,18,19,15)(14,20,16,22)(17,25,26,23)(21,27,24,28)Polycyclic presentation:
2^8|b^2=f,c^2=h,e^2=f,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 | #497519, #6542416, #6591509, … |
| maximal normal subgroup of | #497519, #6542416, #6591509, … |
| maximal quotient of | #521660, #522291, #522792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31017);