Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,11,8,7,10,12);(2,4)(6,7)(11,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,7,9,12,8,6,10,11)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^b=ch,d^c=di152 conjugacy classes — view the full character table.
| maximal subgroup of | #72343941, #72343945, #72343951, … |
| maximal normal subgroup of | #72343941, #72343945, #72343951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89132);