Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
834 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)(17,18,21,26)(19,28,30,23)(20,25,32,39)(22,24,34,27)(29,41,42,36)(31,37,44,35)(33,38,45,40)(43,47,48,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,30)(18,23,26,28)(20,29,32,42)(22,31,34,44)(24,35,27,37)(25,36,39,41)(33,43,45,48)(38,46,40,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(22,34)(23,35)(25,38)(26,27)(28,37)(31,44)(33,45)(36,46)(39,40)(41,47)(43,48);(3,11)(6,9)(10,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,36)(24,38)(26,39)(27,40)(28,41)(30,42)(31,43)(34,45)(35,46)(37,47)(44,48);(2,8)(6,9)(7,14)(13,15)(18,26)(19,30)(20,32)(24,27)(31,44)(33,45)(36,41)(46,47)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #244607, #244611, #244615, … |
| maximal normal subgroup of | #244607, #244611, #244615, … |
| maximal quotient of | #6386193, #6387949, #6388216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54794);