Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#496097 (order 1024) · Inn(G) = C22 · Out(G) = C2×#2929 |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C32 | C22 |
| commutator | C2 | C4×C32 |
| socle, prime radical | C22 | C2×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 68 points:
(5,6,7,13,8,14,17,31,9,15,18,32,21,34,37,53,10,16,19,33,22,35,38,54,24,36,40,55,43,56,57,66,11,12,20,27,23,28,39,47,25,29,41,48,44,50,58,62,26,30,42,49,45,51,59,63,46,52,60,64,61,65,67,68);(1,2,3,4)(6,12)(13,27)(14,28)(15,29)(16,30)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(53,62)(54,63)(55,64)(56,65)(66,68)Polycyclic presentation:
2^8|a^2=c,b^2=d,c^2=e,e^2=f,f^2=g,g^2=h,b^a=bh160 conjugacy classes — view the full character table.
| Frattini subgroup of | #72230722, #72230728, #72230767, … |
| maximal subgroup of | #94505, #94508, #94604, … |
| maximal normal subgroup of | #94505, #94508, #94604, … |
| maximal quotient of | #94504, #94507, #94755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 498);