Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(10,12)(14,18)(15,21)(20,25)(22,27);(2,4)(6,9)(7,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^8|b^2=f,c^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37779, #7635000, #7635100, … |
| maximal normal subgroup of | C2×#37779, #7635000, #7635100, … |
| maximal quotient of | #521330, #522072, #522578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30767);