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) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2431 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1143 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)(5,6,9,7)(8,11,12,10)(13,14)(15,21)(16,18)(17,24)(19,23)(20,25)(22,26)(27,28);(6,7)(10,11)(14,16)(19,21)(20,24)(26,27);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(5,8,9,12)(6,10,7,11)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=f,e^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 | #214044, #214642, #6481486, … |
| maximal normal subgroup of | #214044, #214642, #6481486, … |
| maximal quotient of | #553762, #553769, #554023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30379);