Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,23)(10,13)(11,18)(14,24)(16,31)(17,27)(19,26)(20,34)(21,35)(22,29)(25,33)(28,32)(30,36);(5,7,10,19)(6,11,13,27)(8,20,22,16)(9,17,24,18)(12,28,29,25)(14,26,23,15)(21,33,36,32)(30,31,35,34);(2,4)(5,8,10,22)(6,12,13,29)(7,16,19,20)(9,21,24,36)(11,25,27,28)(14,30,23,35)(15,31,26,34)(17,32,18,33);(7,17)(9,24)(11,26)(14,23)(15,27)(16,32)(18,19)(20,33)(21,36)(25,34)(28,31)(30,35);(6,13)(11,27)(12,29)(14,23)(15,26)(25,28)(30,35)(31,34)Polycyclic presentation:
2^8|b^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497331, #6573737, #6792446, … |
| maximal normal subgroup of | #497331, #6573737, #6792446, … |
| maximal quotient of | #6358588, #6358595, #6358603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54048);