Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,15)(11,14,12,16)(17,18,20,23)(19,25,27,22)(21,24,29,31)(26,30,28,32);(2,5)(3,8)(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,32)(28,29)(30,31);(9,12,13,11)(10,14,15,16)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,3,4,8)(2,6,5,7)(10,15)(14,16)(18,23)(19,27)(24,31)(26,28)Polycyclic presentation:
2^8|a^2=efh,b^2=fh,c^2=fh,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6642942, #6642952, #6642970, … |
| maximal normal subgroup of | #6642942, #6642952, #6642970, … |
| maximal quotient of | #384644, #384738, #384802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25491);