Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17,21,28)(8,22,25,30)(9,15)(10,16)(11,14)(12,19,27,20)(13,24,29,23)(18,32)(26,35)(31,33)(34,36);(1,2,3,4)(5,7,9,20,11,21,10,19)(6,12,15,28,14,27,16,17)(8,18,24,33,25,35,23,34)(13,26,22,36,29,32,30,31);(5,8)(6,13)(7,18)(9,24)(10,23)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,34)(20,33)(21,35)(27,32)(28,36);(6,14)(12,27)(13,29)(15,16)(17,28)(22,30)(26,32)(31,36)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #137672, #359578, #359686, … |
| maximal normal subgroup of | #137672, #359578, #359686, … |
| maximal quotient of | #193537, #193538, #193539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13905);