Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,20)(17,24)(18,23)(19,22)(21,27)(26,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,16)(14,20,23,25)(17,27,19,26)(21,24,28,22);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,21,23,28)(17,27,19,26)(20,24,25,22);(6,9)(10,12)(13,17,18,19)(14,22,23,24)(15,26,16,27)(20,28,25,21)Polycyclic presentation:
2^8|b^2=h,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24017, #6440313, #6440319, … |
| maximal normal subgroup of | C2×#24017, #6440313, #6440319, … |
| maximal quotient of | #254713, #254720, #254727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17005);