Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C22 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,12,17,13,18,24,32,14,16,25,29,26,30,37,39)(11,19,21,31,22,33,35,40,20,15,23,27,34,28,36,38);(1,2,3,5,4,6,7,8)(9,11)(10,15)(12,23)(13,22)(14,20)(16,19)(17,31)(18,28)(21,25)(24,36)(26,34)(27,29)(30,33)(32,40)(35,37)(38,39);(11,20)(15,19)(21,23)(22,34)(27,31)(28,33)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=di,c^b=ci,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72224829, #72224832, #72224833, … |
| maximal normal subgroup of | #72224829, #72224832, #72224833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33751);