Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,27)(22,36)(23,28)(25,32)(29,38)(30,39)(31,41)(33,42)(34,40)(35,43)(37,45)(44,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,23,31)(18,24,28,41)(20,29,33,46)(21,34,22,30)(25,37,43,47)(26,38,42,48)(27,40,36,39)(32,45,35,44);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,32)(22,35)(23,33)(24,37)(26,36)(27,42)(28,43)(30,44)(31,46)(34,45)(38,39)(40,48)(41,47);(3,11)(6,9)(10,16)(13,15)(17,20)(18,26)(19,29)(21,35)(22,32)(23,33)(24,38)(25,36)(27,43)(28,42)(30,45)(31,46)(34,44)(37,39)(40,47)(41,48);(2,8)(6,9)(7,14)(13,15)(17,21,23,22)(18,27,28,36)(19,30,31,34)(20,32,33,35)(24,39,41,40)(25,42,43,26)(29,44,46,45)(37,38,47,48)Polycyclic presentation:
2^8|b^2=h,c^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61790, #10577968, #10578022, … |
| maximal normal subgroup of | C2×#61790, #10577968, #10578022, … |
| maximal quotient of | #6386185, #6387872, #6387873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54778);