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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,16)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32);(2,6)(4,7)(10,13)(11,15)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32);(1,2)(3,7)(4,8)(5,6)(9,10,12,13)(11,14,15,16)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32);(1,3,5,8)(2,7,6,4)(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32);(1,4,5,7)(2,8,6,3)(9,11)(10,14)(12,15)(13,16)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fg,d^2=fgh,e^2=gh,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×#40435, #7637716, #7638139, … |
| maximal normal subgroup of | C2×#40435, #7637716, #7638139, … |
| maximal quotient of | #522028, #522545, #523180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33423);