Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481349 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #60695 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
796 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)(9,10,13,18,14,20,29,19)(11,21,24,36,22,16,32,35)(12,25,27,17,28,15,26,31)(23,30,39,37,38,33,40,34);(1,3)(2,5)(4,6)(7,8)(10,15)(11,22)(12,26)(13,29)(16,33)(17,18)(19,31)(20,25)(21,30)(23,39)(27,28)(34,36)(35,37)(38,40);(2,3)(6,7)(9,11)(10,16)(12,23)(13,24)(14,22)(15,30)(17,34)(18,35)(19,36)(20,21)(25,33)(26,40)(27,39)(28,38)(29,32)(31,37);(11,22)(16,21)(23,38)(24,32)(30,33)(34,37)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=f,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216730, #72216916, #72216926, … |
| maximal normal subgroup of | #72216730, #72216916, #72216926, … |
| maximal quotient of | C2×#481349, #72318476, #72318557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388545);