Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#386 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,15,24)(4,16)(5,11)(6,20)(7,10)(8,14,23,19)(9,17)(13,28,31,22)(18,26)(21,25)(27,29,32,30);(1,3,5,19,7,15,6,14)(2,8,11,24,10,23,20,12)(4,13,17,30,18,31,21,29)(9,22,26,32,25,28,16,27);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,29)(15,31)(16,20)(19,30)(23,28)(24,32);(2,10)(8,23)(9,25)(11,20)(12,24)(16,26)(22,28)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,g^b=gh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #359575, #359619, #359620, … |
| maximal normal subgroup of | #359575, #359619, #359620, … |
| maximal quotient of | #193727, #193728, #193729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13927);