Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,10)(11,12)(14,15)(16,23)(20,22)(24,28);(2,4)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(5,7,9,8)(6,11,10,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=fg,e^2=fg,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 | C2×#37830, #7635000, #7635127, … |
| maximal normal subgroup of | C2×#37830, #7635000, #7635127, … |
| maximal quotient of | #521390, #522118, #522614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30818);