Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| 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×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1067 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)(5,6,8,10)(7,9,12,11)(13,14)(15,17)(16,24)(18,19)(20,23)(21,26)(22,27)(25,28);(5,6,8,10)(7,11,12,9)(14,19)(17,20)(21,24)(25,27);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,22)(17,19)(18,23)(21,27)(24,25)(26,28);(13,16,18,26)(14,21,19,24)(15,22,23,28)(17,25,20,27);(5,7,8,12)(6,9,10,11)(13,17,18,20)(14,15,19,23)(16,25,26,27)(21,22,24,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fgh,d^2=f,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 | #98412, #130566, #130568, … |
| maximal normal subgroup of | #98412, #130566, #130568, … |
| maximal quotient of | #521493, #522145, #522651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32734);