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 | |
1164 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)(6,8)(10,11)(13,14,18,20)(15,21,22,17)(16,19,26,28)(23,27,24,25);(5,6,9,8)(7,11,12,10)(15,22)(17,21)(23,24)(25,27);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15)(14,17)(16,24)(18,22)(19,27)(20,21)(23,26)(25,28);(5,8,9,6)(7,11,12,10)(13,16)(14,19)(15,23)(17,25)(18,26)(20,28)(21,27)(22,24);(13,17)(14,15)(16,25)(18,21)(19,23)(20,22)(24,28)(26,27)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,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 | #214772, #6352407, #6352408, … |
| maximal normal subgroup of | #214772, #6352407, #6352408, … |
| maximal quotient of | #544433, #545017, #545379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34363);