Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| 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 | C24 | #102 |
| 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 | |
805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,8,6)(9,10)(11,20,23,37)(12,16)(13,27)(14,17)(15,22,31,38)(18,25)(19,26)(21,36,24,34)(28,35,39,30)(29,32)(33,40);(1,3)(2,5)(4,6)(7,8)(9,11,14,23)(10,15,17,31)(12,21,26,24)(13,28,29,39)(16,30,19,35)(18,34,33,36)(20,27,37,32)(22,25,38,40);(2,3)(6,7)(9,12,14,26)(10,16,17,19)(11,21,23,24)(13,25,29,40)(15,30,31,35)(18,32,33,27)(20,36,37,34)(22,28,38,39);(10,17)(15,31)(16,19)(18,33)(20,37)(27,32)(30,35)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388542);