Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C16×S4 (order 768) · Inn(G) = C22 · Out(G) = C2×D3×C16 |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C64 | C22 |
| commutator, socle | C2 | C22×C32 |
| Frattini, cosocle | C32 | C23 |
| prime radical | #41 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C64 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,4,10,5,11,20,35,6,12,21,36,24,39,55,75,7,13,22,37,25,40,56,76,27,42,58,78,61,81,95,110,8,14,23,38,26,41,57,77,28,43,59,79,62,82,96,111,29,44,60,80,63,83,97,112,64,84,98,113,99,114,120,126)(3,9,16,31,17,32,49,69,18,33,50,70,52,72,91,106,19,34,51,71,53,73,92,107,54,74,93,108,94,109,119,125,15,30,45,65,46,66,85,100,47,67,86,101,88,103,115,121,48,68,87,102,89,104,116,122,90,105,117,123,118,124,127,128);(1,3)(2,9)(4,16)(5,17)(6,18)(7,19)(8,15)(10,31)(11,32)(12,33)(13,34)(14,30)(20,49)(21,50)(22,51)(23,45)(24,52)(25,53)(26,46)(27,54)(28,47)(29,48)(35,69)(36,70)(37,71)(38,65)(39,72)(40,73)(41,66)(42,74)(43,67)(44,68)(55,91)(56,92)(57,85)(58,93)(59,86)(60,87)(61,94)(62,88)(63,89)(64,90)(75,106)(76,107)(77,100)(78,108)(79,101)(80,102)(81,109)(82,103)(83,104)(84,105)(95,119)(96,115)(97,116)(98,117)(99,118)(110,125)(111,121)(112,122)(113,123)(114,124)(120,127)(126,128);(3,15)(9,30)(16,45)(17,46)(18,47)(19,48)(31,65)(32,66)(33,67)(34,68)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(91,115)(92,116)(93,117)(94,118)(106,121)(107,122)(108,123)(109,124)(119,127)(125,128)Polycyclic presentation:
2^8|a^2=d,d^2=e,e^2=f,f^2=g,g^2=h,c^b=ch160 conjugacy classes — view the full character table.
| Fitting subgroup of | #13737⋊C3 |
| maximal subgroup of | #94818, #153586, #153587, … |
| maximal normal subgroup of | #94818, #153586, #153587, … |
| maximal quotient of | #153423, #153428, #153568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6725);