Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11915 |
| automorphism | Aut(G) = #72306331 (order 2048) · Inn(G) = #1533 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1533 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #97 | C23 |
| prime radical | #532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,4,10,17,28,6,11,21,29,18,8,20,26)(3,12,14,30,15,7,22,24,16,23,31,27,13,25,19,32);(1,3)(2,7)(4,13)(5,19)(6,16)(8,23)(9,27)(10,12)(11,25)(14,20)(15,18)(17,31)(21,22)(24,28)(26,32)(29,30);(1,4,6,18)(2,8,11,10)(3,13,16,15)(5,20,21,17)(7,23,25,12)(9,28,29,26)(14,22,31,19)(24,30,32,27)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=g,e^2=f,f^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,e^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94771, #94773, #133263, … |
| maximal normal subgroup of | #94771, #94773, #133263, … |
| maximal quotient of | #124511, #124547, #124620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4905);