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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
801 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(7,8)(9,10,14,11)(12,15,13,16)(17,18,21,19)(20,23,24,22);(9,11,14,10)(12,15,13,16)(18,19)(22,23);(1,2)(3,7)(4,8)(5,6)(9,12,14,13)(10,15,11,16)(17,19,21,18)(20,23,24,22);(1,3,5,4)(2,7,6,8)(17,20)(18,22)(19,23)(21,24);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=h,d^2=g,e^2=fgh,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×#39835, #7637383, #7637411, … |
| maximal normal subgroup of | C2×#39835, #7637383, #7637411, … |
| maximal quotient of | #521526, #522174, #522672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32823);