Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,16,20)(15,22,23,18)(17,19,25,21)(24,27,28,26);(1,2,4,3)(5,7,8,11)(6,9,10,12);(6,10)(9,12)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,26,21,27);(1,3,4,2)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28)Polycyclic presentation:
2^8|a^2=h,b^2=efh,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #100248, #100250, #100264, … |
| maximal normal subgroup of | #100248, #100250, #100264, … |
| maximal quotient of | #351297, #351298, #351299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26102);