Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20,23,37)(12,16)(13,27)(14,19)(15,22,31,39)(17,25)(18,26)(21,36,24,33)(28,35,40,30)(29,32)(34,38);(9,11)(10,15)(12,21)(13,28)(14,23)(16,30)(17,33)(18,35)(19,31)(20,32)(22,38)(24,26)(25,39)(27,37)(29,40)(34,36);(2,5)(6,8);(9,12,14,26)(10,16,19,18)(11,21,23,24)(13,25,29,38)(15,30,31,35)(17,32,34,27)(20,36,37,33)(22,28,39,40)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285482, #72285579, #72285656, … |
| maximal normal subgroup of | #72285482, #72285579, #72285656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388470);