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 |
| prime radical | #3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(17,18,23,28)(19,24,32,42)(20,26,34,44)(21,27,22,37)(25,36,45,33)(29,46,30,39)(31,40,35,41)(38,47,43,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,30)(21,35)(22,31)(23,32)(25,43)(26,39)(27,41)(28,42)(29,34)(33,47)(36,48)(37,40)(38,45)(44,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,38)(26,37)(27,44)(28,45)(30,32)(31,47)(35,48)(39,41)(40,46)(42,43);(3,11)(6,9)(10,16)(13,15)(17,20)(18,26)(19,30)(21,36)(22,33)(23,34)(24,39)(25,37)(27,45)(28,44)(29,32)(31,47)(35,48)(38,41)(40,43)(42,46);(2,8)(6,9)(7,14)(13,15)(17,21,23,22)(18,27,28,37)(19,31,32,35)(20,33,34,36)(24,40,42,41)(25,44,45,26)(29,47,30,48)(38,46,43,39)Polycyclic presentation:
2^8|a^2=h,c^2=f,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 | #249810, #249830, #249833, … |
| maximal normal subgroup of | #249810, #249830, #249833, … |
| maximal quotient of | #6386188, #6387882, #6388240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54786);