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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
999 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)(5,6,9,8)(7,10,12,11)(13,14)(15,20)(16,24)(17,22)(18,19)(21,26)(23,25)(27,28);(5,7)(6,10)(8,11)(9,12)(14,19)(15,22)(21,24)(23,28);(1,2,3,4)(5,6,9,8)(7,11,12,10)(13,15,18,22)(14,20,19,17)(16,23,26,28)(21,27,24,25);(13,16,18,26)(14,21,19,24)(15,23,22,28)(17,25,20,27);(5,8,9,6)(7,10,12,11)(13,17,18,20)(14,22,19,15)(16,25,26,27)(21,28,24,23)Polycyclic presentation:
2^8|a^2=fh,c^2=gh,d^2=f,e^2=h,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 | #98033, #166329, #166330, … |
| maximal normal subgroup of | #98033, #166329, #166330, … |
| maximal quotient of | #521428, #522194, #522601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32472);