Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,21,17,20,26,28)(15,16,23,25,22,24,19,27);(6,9)(10,12)(13,15,18,23,17,22,26,19)(14,16,21,25,20,24,28,27);(2,4)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27);(5,7,8,11)(6,10,9,12)(14,20)(16,24)(21,28)(25,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21110, #6371728, #6371733, … |
| maximal normal subgroup of | C2×#21110, #6371728, #6371733, … |
| maximal quotient of | #203400, #203401, #203402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14098);