Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #2184 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,21,28,23,26,22,39)(19,29,32,43,30,46,33,25)(20,34,35,44,38,24,37,27)(31,45,48,41,36,42,47,40);(3,9)(6,12)(10,15)(13,16)(18,24)(19,30)(20,35)(21,22)(25,41)(26,34)(27,39)(28,44)(29,45)(31,47)(36,48)(37,38)(40,43)(42,46);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25,26,43)(20,36)(21,32)(22,33)(23,30)(24,40,34,41)(27,45,44,42)(28,29,39,46)(31,38)(35,47)(37,48);(2,7)(6,12)(8,14)(13,16)(18,26)(19,30)(24,34)(27,44)(28,39)(31,36)(32,33)(47,48)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286391, #72286506, #72286518, … |
| maximal normal subgroup of | #72286391, #72286506, #72286518, … |
| maximal quotient of | #72343329, #72343373, #72343448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391124);