Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
757 conjugacy classes of subgroups; only the 423 normal subgroups are shown.
This group has 757 conjugacy classes of subgroups — too many to draw. Only its 423 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(10,13)(14,15)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30);(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(19,25)(21,29)(23,27)(24,28);(1,3,5,2)(4,7,8,6)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,4,5,8)(2,6,3,7)(9,11,12,16)(10,14,13,15)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32);(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130529, #132757, #253931, … |
| maximal normal subgroup of | #130529, #132757, #253931, … |
| maximal quotient of | #535421, #536134, #536541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38424);