Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | #2404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(16,24)(19,26)(22,28)(25,27);(6,9)(10,11)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)Polycyclic presentation:
2^8|b^2=fh,d^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132038, #132150, #213924, … |
| maximal normal subgroup of | #132038, #132150, #213924, … |
| maximal quotient of | #553716, #553978, #554166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30174);