Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C23×#1202 (order 8192) · Inn(G) = C25 · Out(G) = C23×H(5,𝔽2) |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | D4×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16,12,14,20,21)(11,17,19,24,18,15,22,23);(1,3)(2,5)(4,8)(6,7)(10,14)(15,17)(16,21)(23,24);(3,8)(5,7)(10,14)(15,17)(16,21)(23,24);(2,6)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24);(11,18)(15,17)(19,22)(23,24)Polycyclic presentation:
2^8|a^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cfh,e^d=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129689, #210268, #210269, … |
| maximal normal subgroup of | #129689, #210268, #210269, … |
| maximal quotient of | #6370183, #6370226, #6370298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53229);