Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1120 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)(5,6,9,12)(7,8,11,10)(14,18)(15,21)(20,23)(22,27);(7,11)(8,10)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26);(1,2)(3,4)(5,7,9,11)(6,10,12,8);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(5,8,9,10)(6,11,12,7)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,e^2=fh,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 | #130481, #248541, #248636, … |
| maximal normal subgroup of | #130481, #248541, #248636, … |
| maximal quotient of | #521403, #522177, #522578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32453);