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 | C2×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1047 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)(5,6,9,11)(7,8,10,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26);(5,7)(6,8)(9,10)(11,12)(14,15)(19,21)(20,22)(26,27);(1,2,3,4)(5,8)(6,10)(7,11)(9,12)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26);(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,8)(6,10)(7,11)(9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=f,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 | #480999, #481001, #6542398, … |
| maximal normal subgroup of | #480999, #481001, #6542398, … |
| maximal quotient of | #521356, #522090, #522701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30858);