Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14,17,15)(16,18,20,19)(23,27)(25,28);(7,11)(10,12)(14,15)(16,20)(21,22,24,26)(23,28,27,25);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,18,20,19)(21,23,24,27)(22,25,26,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,20)(14,18,15,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=h,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #136167, #136243, #137347, … |
| maximal normal subgroup of | #136167, #136243, #137347, … |
| maximal quotient of | #154057, #154058, #154067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7627);