Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
739 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)(4,5)(6,8)(9,10,13,16)(11,14,12,15)(17,18,22,25)(19,20,23,27)(21,24,29,31)(26,28,30,32);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(19,23)(20,27)(21,29)(24,31);(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,26)(22,23)(24,28)(25,27)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,3)(2,6)(4,8)(5,7)(17,21,22,29)(18,24,25,31)(19,26,23,30)(20,28,27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51836, #7707697, #7707852, … |
| maximal normal subgroup of | C2×#51836, #7707697, #7707852, … |
| maximal quotient of | #561271, #561284, #562436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44824);