Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(9,12);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(14,18)(19,25)(20,26)(27,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(13,15,17,22)(14,19,18,25)(16,23,24,21)(20,28,26,27);(13,16,17,24)(14,20,18,26)(15,21,22,23)(19,27,25,28)Polycyclic presentation:
2^8|b^2=fg,c^2=fg,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #35983⋊C3 |
| maximal subgroup of | #132352, #248678, #248679, … |
| maximal normal subgroup of | #132352, #248678, #248679, … |
| maximal quotient of | #517811, #517835, #517845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28971);