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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1057 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)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(2,3)(14,18)(19,21)(20,25)(26,27);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(1,3,4,2)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|c^2=fh,d^2=fg,e^2=h,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×#39394, #7636633, #7636678, … |
| maximal normal subgroup of | C2×#39394, #7636633, #7636678, … |
| maximal quotient of | #521450, #522164, #522723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32382);