Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#107 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28)(20,25)(21,34)(22,37)(23,27)(24,31)(26,35)(29,41)(30,42)(32,48)(33,44)(36,38)(39,45)(40,47)(43,46);(17,19,29,45,22,32,21,31,23,33,35,46,20,30,36,47)(18,24,38,48,25,39,26,28,27,40,34,42,37,43,41,44);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,22)(18,25,27,37)(19,30,33,32)(21,29,36,35)(24,39,40,43)(26,34,41,38)(28,42,44,48)(31,45,47,46);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=egi,c^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,d^c=dh110 conjugacy classes — view the full character table.
| maximal subgroup of | #72234336, #72234349, #72234351, … |
| maximal normal subgroup of | #72234336, #72234349, #72234351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390667);