Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1114 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)(13,14,19,23)(15,22,17,20)(16,21,26,18)(24,28,27,25);(5,6)(7,11)(8,9)(10,12)(13,15,19,17)(14,20,23,22)(16,24,26,27)(18,28,21,25);(1,2)(3,4)(6,9)(10,11)(13,16)(14,21)(15,24)(17,27)(18,23)(19,26)(20,25)(22,28);(5,6)(7,10)(8,9)(11,12)(13,17,19,15)(14,22,23,20)(16,24,26,27)(18,28,21,25);(5,7,8,12)(6,10,9,11)(13,18,19,21)(14,16,23,26)(15,25,17,28)(20,27,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,e^2=fh,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 | #133242, #214936, #247810, … |
| maximal normal subgroup of | #133242, #214936, #247810, … |
| maximal quotient of | #544449, #545031, #545391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35415);