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 |
| prime radical | #2699 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1045 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)(13,14,19,18)(15,22,16,20)(17,25,26,21)(23,27,24,28);(2,4)(13,15,19,16)(14,20,18,22)(17,23,26,24)(21,27,25,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,20,18,22)(17,24,26,23)(21,27,25,28);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(1,2)(3,4)(13,18,19,14)(15,20,16,22)(17,25,26,21)(23,27,24,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=g,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×#37735, #7635106, #7635115, … |
| maximal normal subgroup of | C2×#37735, #7635106, #7635115, … |
| maximal quotient of | #521370, #522103, #522610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30723);