Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1190 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)(6,9)(7,11)(13,14,18,22)(15,23,24,19)(16,20,26,28)(17,21,27,25);(5,6,8,9)(7,10,11,12)(15,24)(17,27)(19,23)(21,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,22,23)(16,25,26,21)(17,20,27,28);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,21)(17,19)(18,26)(22,28)(23,27)(24,25);(13,17)(14,21)(15,20)(16,19)(18,27)(22,25)(23,26)(24,28)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #482902, #6353262, #6353272, … |
| maximal normal subgroup of | #482902, #6353262, #6353272, … |
| maximal quotient of | #544523, #545082, #545507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34401);