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 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(11,15)(14,16)(17,18,21,23)(19,20,24,22);(2,5)(6,8)(9,10,13,12)(11,14,15,16)(17,19,21,24)(18,22,23,20);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19)(18,20)(21,24)(22,23);(1,3,4,7)(2,6,5,8)(9,10,13,12)(11,14,15,16)(17,20)(18,19)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=g,d^2=fgh,e^2=g,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×#38879, #7635991, #7636326, … |
| maximal normal subgroup of | C2×#38879, #7635991, #7636326, … |
| maximal quotient of | #521858, #522425, #523106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31867);