Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,24)(19,27)(21,23)(25,32)(28,29)(30,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,22,24)(19,28,30,25)(20,23,26,21)(27,32,31,29);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,22,30)(18,25,24,28)(20,29,26,32)(21,27,23,31);(3,11)(6,9)(10,16)(13,15)(17,20,22,26)(18,21,24,23)(19,29,30,32)(25,27,28,31);(2,8)(6,9)(7,14)(13,15)(17,21,22,23)(18,26,24,20)(19,27,30,31)(25,32,28,29)Polycyclic presentation:
2^8|b^2=h,c^2=f,d^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #249811, #249812, #249831, … |
| maximal normal subgroup of | #249811, #249812, #249831, … |
| maximal quotient of | #6386188, #6387880, #6387881, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54788);