Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1028 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)(5,6)(7,8)(9,11)(10,12)(13,14,18,20)(15,17,21,26)(16,19,25,28)(22,24,23,27);(5,7,10,9)(6,11,12,8)(15,21)(17,26)(22,23)(24,27);(1,2,3,4)(13,15)(14,17)(16,23)(18,21)(19,27)(20,26)(22,25)(24,28);(5,8,10,11)(6,7,12,9)(13,16)(14,19)(15,22)(17,24)(18,25)(20,28)(21,23)(26,27);(5,9,10,7)(6,11,12,8)(13,17)(14,15)(16,24)(18,26)(19,22)(20,21)(23,28)(25,27)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#43544, #7667859, #7668093, … |
| maximal normal subgroup of | C2×#43544, #7667859, #7668093, … |
| maximal quotient of | #535432, #535766, #536102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36532);