Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,20)(11,18,24,16)(12,25,28,36)(13,21,15,19)(17,30,34,27)(22,37,26,40)(23,32,29,35)(31,39,33,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,24)(10,16,20,18)(12,26,28,22)(13,29,15,23)(17,33,34,31)(19,35,21,32)(25,37,36,40)(27,38,30,39);(9,12,14,28)(10,17,20,34)(11,22,24,26)(13,27,15,30)(16,31,18,33)(19,25,21,36)(23,38,29,39)(32,37,35,40);(3,8)(5,7)(9,11)(10,18)(12,22)(13,29)(14,24)(15,23)(16,20)(17,33)(19,32)(21,35)(25,37)(26,28)(27,39)(30,38)(31,34)(36,40);(9,13,14,15)(10,19,20,21)(11,23,24,29)(12,27,28,30)(16,32,18,35)(17,25,34,36)(22,38,26,39)(31,37,33,40)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6575781, #6575785, #6575787, … |
| maximal normal subgroup of | #6575781, #6575785, #6575787, … |
| maximal quotient of | #6379104, #6379330, #6379726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54212);