Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(10,12)(13,14,18,22)(15,20,16,19)(17,21,25,28)(23,27,24,26);(2,4)(13,15,18,16)(14,19,22,20)(17,24,25,23)(21,27,28,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,20,22,19)(17,24,25,23)(21,27,28,26);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(1,2)(3,4)(13,17,18,25)(14,21,22,28)(15,23,16,24)(19,26,20,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37743, #7635106, #7635130, … |
| maximal normal subgroup of | C2×#37743, #7635106, #7635130, … |
| maximal quotient of | #521378, #522108, #522618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30731);