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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10)(11,15)(12,14)(13,16);(1,2,5,3)(4,7,8,6)(9,10,12,14)(11,13,16,15)(18,21)(22,24);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,20,23)(18,22,21,24);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^8|b^2=f,d^2=fgh,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×#38510, #7635970, #7636313, … |
| maximal normal subgroup of | C2×#38510, #7635970, #7636313, … |
| maximal quotient of | #521832, #522414, #523088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31498);