Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C42 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(13,14,17,22,18,24,32,41,19,23,33,40,16,21,30,38)(15,25,28,42,29,20,34,36,26,37,31,44,27,35,43,39);(2,4)(5,6,7,9,8,10,11,12)(13,15)(14,20,23,35)(16,27)(17,31)(18,29)(19,26)(21,25,24,37)(22,39,40,36)(28,33)(30,34)(32,43)(38,44,41,42);(15,26)(20,35)(25,37)(27,29)(28,31)(34,43)(36,39)(42,44)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,c^b=ci,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321356, #72321363, #72321367, … |
| maximal normal subgroup of | #72321356, #72321363, #72321367, … |
| maximal quotient of | #72322486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33747);