Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(13,14,18,19)(15,22,20,16)(17,21,25,26)(23,28,27,24);(2,4)(5,7,9,6)(8,11,12,10)(14,19)(15,20)(21,26)(23,27);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,15,18,20)(14,16,19,22)(17,23,25,27)(21,24,26,28);(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,26)(25,28);(1,2)(3,4)(13,17,18,25)(14,21,19,26)(15,23,20,27)(16,24,22,28)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=g,e^2=f,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 | #97816, #246037, #246038, … |
| maximal normal subgroup of | #97816, #246037, #246038, … |
| maximal quotient of | #521361, #522088, #522593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30696);