Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483882 |
| 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 | |
443 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)(3,6,9,13,10,12,15,16)(17,18,21,28,23,26,22,39)(19,29,32,42,30,45,33,25)(20,34,35,43,38,24,37,27)(31,44,46,48,36,41,47,40);(18,24)(20,35)(21,22)(25,40)(26,34)(27,39)(28,43)(29,41)(31,46)(32,33)(36,47)(37,38)(42,48)(44,45);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25,26,42)(20,36)(21,32)(22,33)(23,30)(24,40,34,48)(27,44,43,41)(28,29,39,45)(31,38)(35,47)(37,46);(2,7)(6,12)(8,14)(13,16)(18,26)(19,30)(24,34)(27,43)(28,39)(31,36)(32,33)(46,47)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,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216788, #72216810, #72216824, … |
| maximal normal subgroup of | #72216788, #72216810, #72216824, … |
| maximal quotient of | C2×#483882, #72318592, #72318619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391078);